Control of a Robotic Leg for Walking on Irregular Surfaces

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Control of a Robotic Leg for Walking on Irregular Surfaces

Nuno Pereira Azevedo Wallenstein Teixeira

Thesis for the Master of Science Degree in

Engineering Physics

Supervisor(s): Prof. Doutor Rui Manuel Agostinho Dilão

Examination Committee

Chairperson: Prof. Doutor Ilídio Pereira Lopes Supervisor: Prof. Doutor Rui Manuel Agostinho Dilão

Members of the Committee: Prof. Doutora Ana Maria Ribeiro Ferreira Nunes

July 2020

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Anyone who has never made a mistake has never tried anything new.

Albert Einstein

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Agradecimentos

Gostaria de agradecer ao professor Rui Dilão do Instituto Superior Técnico pela oportunidade de desenvolver, ajudar-me nesta tese e por ter acreditado em mim.

Gostaria de agradecer à minha família, nomeadamente aos meus pais e irmã por todo o apoio e sacrifício depositado em mim para eu completar o curso. Em especial, queria agradecer à minha avó por me ajudar a ser melhor pessoa e me dar oportunidade de crescer e à minha tia Clara por toda a disponibilidade enquanto estive em Lisboa.

Por último gostaria de te agradecer Joana, por me ajudares a perceber que também há pessoas como eu.

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Resumo

Apresentamos um modelo de locomoção articulado baseado no pêndulo invertido, em que as pernas não têm massa, capaz de andar em superfícies irregulares. O passo do robot é composto pelo movimento pendular da perna que o suporta (perna de suporte), e, o movimento articulado associado às juntas do joelho e da anca, dividido em 3 fases, que possibilita o robot passar obstáculos (perna articulada). A adaptação para superfícies irregulares foi possível controlando o tempo do passo de modo às duas pernas fazerem sempre o mesmo ângulo quando um passo começa. Adicionalmente, conservamos a energia mecânica do robot de modo a este continuar a andar com sucesso preservando as mesmas características do passo anterior. Formulamos dois critérios de falha relativamente ao regime de locomoção e à superfície em questão: um deles para os máximos locais, e outro relativo aos mínimos locais. A junção destes dois critérios permitiu formular uma condição suficiente de modo o robot andar sem falhar o passo devido à perna de suporte. Para a perna articulada as condições de falhanço foram defininas mas não formulamos condições suficientes. Neste modelo estabelecemos uma relação entre a energia necessária para manter uma trajectória de ciclo limite, o ângulo da rampa e a abertura de pernas quando o robot está a descer um plano inclinado sem controlo activo para conservar a energia mecânica no começo de cada passo.

Palavras Chave

perna de suporte, perna articulada, modelos de locomoção, superfícies irregulares, ciclo limite

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Abstract

We present an articulated walking model based on the inverted pendulum, with massless rigid legs, capable of walking on irregular surfaces. The step of the biped robot is composed by the pendular motion of the leg that supports the robot (standing leg), and, an articulated motion associated to the knee and the hip joints of the other leg, divided in 3 phases, that enables the robot to overcome obstacles (trailing leg). The adaptation to irregular surfaces was possible by controlling the time of the step so that both legs made always the same angle between them when the step starts. Additionally, we conserved the mechanical energy of the robot in order to continue walking successfully with the same characteristics as the step before. We formulated two failing criteria for the robot to walk on irregular surfaces, one for the local maximums and other for the local minimums. The junction of both failing criteria allowed to formulate a global sufficient condition in order for the robot to walk without failing the step due to the standing leg. For the articulated leg the failure conditions were defined but no sufficient conditions were formulated. We also establish a relationship between the energy needed to maintain a limit cycle trajectory, the angle of the slope, and the aperture between the legs when the robot is walking down a ramp without active control present to conserve the mechanical energy at the start of each step.

Keywords

standing leg, trailing leg, walking models, irregular surfaces, limit cycle

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List of Figures

1.1 Running/hopping model described in [1]. This model displays the basic dynamics of running/hopping. It is charecterized by two phases, the ground phase and the aerial phase. The angle which the leg returns to the ground is the same as in the previous step. In this scheme,~vis the velocity,mis the mass,αis the angle of attack andβis the angle the velocity makes with the ground. . . 3 1.2 Bipedal spring-mass walking model described in [2]. The initial velocity is given by~v,

in this case,β=0, this is the velocity is initially parallel to the ground,αTDis the angle of attack for each leg, this is, the aperture that the leg has when it starts the double support phase. . . 5 1.3 Continuous phase described in [3] . . . 6 1.4 Switch phase described in [3] . . . 7 2.1 Transition instant of the two legs between steps. The massm, the foot associated to the

step before and the foot associated to the next step always create an isosceles triangle with the aperture between the legs corresponding to the angleβ. Although the angle βis the parameter of the model, both of the anglesαandβare equivalent since both express the degree of openness between the legs when a step transition occurs. . . 13 2.2 Phase curves for the inverted pendulum in Eq. (2.5) for,m = 80 Kg,l = 1 m g = 9.8

m/s², and for various values of energyH=E, namely,E=500 J<mlg(blue),E=784 J=mlg(orange) andE=900 J>mlg(green). The level of energy ofE=mlg=784 J, is the phase curve associated boundary of the energetic sufficient conditionE = mgl.

Only solutions with ˙φ≤0 are taken. . . 14 2.3 Initial and final states of the standing leg in the first step and its trajectory on a flat

surface with the parameters of Eq. (2.8). Due to Eqs. (2.3) and (2.5), the mass m is propagated in the positive direction of thexxaxis, which means that ˙φ(t_i₍₁₎)<0, from φ0=φ(t_i₍₁₎)toφ(t_f₍₁₎). In the case of a flat horizontal surfaceφ(t_f₍₁₎) =α, however the initial condition,φ0, can be different fromπ−α. Bothφ(t_i₍₁₎)andφ(t_f₍₁₎)are measured from a horizontal line, passing through the foot of the standing leg. . . 16

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2.4 Step 1 of a robot along an horizontal flat surface with a fixed angle apertureβbetween the legs at the initial (t=t_i₍₁₎) and final (t=t_f₍₁₎) instants of the first step. The legs of the robot have lengthl, the standing leg has an the initial angle ofφ₀and the movement of the mass is in the positive direction of thexxaxis. The standing leg and the trailing leg both start its movement at the same time. The standing leg applies a pendular motion to the mass, and the trailing leg is composed of a 2-link articulation mechanism where the hip joint controls the thigh with lengthl1and the knee joint controls the calf with lengthl2in such a way thatl =l1+l2. At the end of the step due to the motion of the trailing leg, a new standing leg with its proper angle of attack is ready to take place. In an horizontal flat surface, this angle of attack isφ(ti₍₂₎) =π−α= (π+β)/2.

The transition from the step before to the next one consists in an instantaneous swap between the standing leg and the trailing leg. . . 16 2.5 Standing leg and the trajectory of the mass in the first and second steps of the move-

ment. All the angles are measured from a horizontal line to the respective state of the leg. The three parameters associated with the standing leg areφ0 = 1.76756 rad, E =900 J andβ=1 rad. The transition is instantaneous from one step to another and the angle between both legs at this point isβ. The angle of attack of the standing leg after the first step, for a flat horizontal surface is going to always beπ−α, whereαis given by Fig. (2.1) and Eq. (2.1). . . 18 2.6 Phase space of the standing leg in flat horizontal surface associated to step 1, Fig. (2.6(a))

and step 2, Fig. (2.6(b)). The labels correspond to the times of the beginning and ending of steps 1 and 2 in the phase space. In both of the steps, the representation of the mass in phase space is limited by phase curve associated toE=mlg=784 J. . . 19 2.7 Angles of the thigh (θ) and the calf (µ) of the trailing leg and of the standing leg (φ).

The angleθ, is within the interval of]−π,π[and is measured at the mass, from the standing leg to the thigh of the trailing leg. This angle represents the aperture between the standing and the trailing legs when the calf is aligned with the thigh,µ= π. The angleµis within the interval of]0, 2π[, and is measured at the knee of the trailing leg, from the thigh to the calf. The thigh of the trailing leg is segmented by two points, the hip,H, and the knee,TK, and its length isl1. The calf of the trailing leg is segmented by the foot,TFand the knee,TK, and its length isl₂. The length of the thigh is controlled by a parameter 0<λ<1. The foot of the standing leg is labeled asSF. . . 20 2.8 Control anglesθ(red) and µ(blue) as a function ofη. The parametersτ_b = 0.2 and

τc=0.6 have a direct relationship with the variableηand the considered phase since in the first step, phasea)goes from 1≤η<1+τ_b, phaseb)goes from 1+τ_b≤η<1+τc

and phasec)goes from 1+τ_c≤η<2. . . 26

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2.9 Beginning (filled model) and ending (dashed model) of the phasea)in the first step.

When the step starts,η =1, phasea)also starts, initially, withθ=−βandµ=π, the hip joint starts the rotation, increasing the value ofθtoβ+e, and at the same time the knee rotates, decreasingµtoγ. . . 27 2.10 Beginning (filled model) and ending (dashed model) of the phaseb)in the first step. In

the beginning of this phase,η=1+τ_b, the hip joint continues its rotation from phase a)and the knee rotates in the opposite direction from phasea)increasing,µfromγto π, aligning the hip with the calf. Notice that in the beginning of phaseb), the calf and the hip start in this phase making an angle ofγ=2π/3rad. . . 27 2.11 Beginning (filled model) and ending (dashed model) of the phasec)in the first step.

In the beginning of this phase, with η = ₁+τc the hip joint and the knee finished the respective rotations, atθ = β+eand µ = π, leaving the trailing leg leg to fall, because it is connected to the mass which is falling due to the pendular motion and also because of the counter-clockwise rotation associated to the thigh in this phase, which is controlled by the parameter e. In the final instant of the step 1,η = 2, the trailing leg becomes aligned making an angle ofβwith the standing leg. . . 28 2.12 Different state transitions throughout the first step. The red and blue dashed lines cor-

respond respectively to the trajectory of the knee and the trajectory of the foot throughout the step. Three dynamic parameters, β, Eand φ₀ are associated with the standing leg and for the trailing leg five dynamic parameters are associated,τ_b,γ,τc,e,λ.

With the parameters considered in Eqs. (2.57) and (2.58), the robot makes an articulated movement of the trailing leg in the first step. When the robot finishes the first step, it is ready to begin another one. . . 28 2.13 Step transition sub phase 1. The step transition is modeled by an instantaneous elastic

collision between the two legs and an inelastic collision between the trailing leg and the floor. The standing leg in the next step loses the radial component associated to the velocity~v_f₍_n₎ =~v(t_f₍_n₎), the velocity of the mass at the final instant of stepn. By losing a component of the velocity, without intervention or a control mechanism, the mass loses kinetic energy. . . 29 2.14 Step transition sub phase 2. A control mechanism is applied to reset the energy of the

standing leg. This is done by increasing the polar velocity of the mass,i.e., increasing the polar velocity associated to the standing leg, byv⁰_(n+1). In the case of a flat horizontal surface,v_i₍_n₊₁₎ = v_f₍_n₎. By conserving the absolute value of the velocity of the standing leg in the specific case of a flat horizontal surface, the energyEassociated to the previous step is conserved. . . 30

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2.15 Plot ofv⁰_(n+1), the velocity added instantly at the start of the step in the direction of

−eˆp(t_i₍_n₊₁₎), as a function of βfrom Eq. (2.72) for three different energies for the following parametersm = 80 Kg, l = 1 m, g = 9.8 m/s². This velocity is 0 for β = 0 since the projectionv_f₍_n₎cosβis maximized and therefore no extra velocity is needed to be added, however, in this situation the step length is also 0 and the duration of the step is also 0. Whenβ = π, the velocity needed to conserve the energy of the step is maximized, this means that more energy is needed, but the length of the step is also maximized. The value ofβ=π/2 sets the limit in which the projection of the velocity from the last step, helps the continuation of the new step or not, sincevf₍_n₎cosπ/2=0. 33 2.16 Case where the aperture angleβbetween both legs isβ > π/2. In this situation, the

collision of the trailing leg with the floor hinders the progression of the next step in the sense that the projection of the velocity at the final instant of stepn has the opposite direction as the velocity of the mass at the initial instant of stepn+1. In this case, more energy is needed to be provided by the control mechanism than the energy associated to the previous step in order to increment a step. . . 34 2.17 Case where the aperture angleβbetween both legs isβ < π/2 in the transition from

stepnto stepn+1. In this situation, the collision of the trailing leg with the floor helps the progression to the next step in the sense that the projection of the velocity at the final instant of stepnhas the same direction as the velocity of the mass at the initial instant of the stepn+1. In this case, less energy is needed to be provided by the control mechanism than the energy associated to the previous step in order to increment a step. 34 2.18 Plot of the energy consumed by the control mechanism in order to preserve the energy

of the step as a function ofβaccording to Eq. (2.78) for three distinct values ofEfor the following parametersm= 80 Kg, l = 1 m,g = 9.8 m/s². Whenβis smaller than π/2 the projection of the final velocity helps the continuation of the movement, there- fore, the remaining part of the kinetic energy is added so that the energy on the next step is conserved. Whenβis bigger thanπ/2 the projection of the final velocity hinders the continuation of the movement, and therefore, the control mechanism has to instantly stop the movement from the last step, and afterwards add the kinetic energy to preserve the energy of the stepE. . . 35 3.1 Final instants of first step of a robot on a horizontal and a non-horizontal floor f(x). In

Fig. (3.1(b)), since the height of the second foot, the foot of the trailing leg, is higher than the first one, the foot of the standing leg, the exit angle in the first step is bigger than the case where the floor is flat and horizontal in Fig. (3.1(a)),φ(t˜_f₍₁₎)>φ(t_f₍₁₎), and so, the time elapsed is shorter, ˜t_f₍₁₎ <t_f₍₁₎. Also in Fig. (3.1(b)), upon transitioning from the first to the second step, similarly to chapter 2, the foot of the trailing leg of at the final instant of step 1 becomes the foot of the standing leg in the next step,(xSF₍₂₎,ySF₍₂₎) = (xTF(^˜tf₍₁₎),yTF(t^˜f₍₁₎)). . . 39

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3.2 Geometry associated to the final instant of the first step on an irregular surface given as a function ofx, f(x). Since each leg has lengthl, the dashed segment, the segment that connects both feet, has length 2lcosα. This distance can be projected to the respective axis given that the corresponding angle isφ(t_f₍₁₎)−α. . . 40 3.3 Height difference between the standing leg in stepnand the standing leg in stepn+1.

The energy of the stepnis determined by measuring the velocity and height of mass mat the final instant of stepn. To determine the energy of stepn+1, the velocity and height are measured in a frame located at the initial instant of stepn+1. If there is no control on the energy received or taken by this height difference, our model may be unable increment a step. Since the heighthi₍_n₊₁₎, from the foot of the standing leg in stepn+1 to the massm, is lower than the heighth_f₍_n₎, from foot of the standing leg in the previous step to the mass, a bigger angular velocity is needed to be added in order for the system to conserve the energyEassociated to the step. . . 42 3.4 Robot walking down the first step on a flat inclined surface. Generally, the angle the

segment that connects both feet at the step transition makes with a horizontal line,σ_(n), is different from the slope of the floor,ψ, but in the case of a flat surface they coincide. . 44 3.5 Projectionv⁰_n+1 as a function of the slope of the floorψin Eq. (3.27) and the angle β

between the legs at the transition instants for three different energies. As the energy E/mlgincreases, the associated valuesv⁰_(n+1)also increase. As in the case of a flat surface,v⁰is strongly related toβ, meaning that the higher theβ, the bigger the projection v⁰_(n+1). When E/mlg = _1, ψ > _{0 and}β > π/2, the projected final velocity is lower because of the height difference, which implies that less velocity is needed to stop the movement of the mass from the previous step, which causesv⁰_n+1to drop slightly. . . . 46 3.6 EnergyE_(n+1)⁰ as a function of the slope the floorψand the angle between the legsβ

in transition instants between steps. This quantity shares the same relationship withβ asv⁰_(n+1)in Fig. (3.5), however, in the caseE/mlg = 1 there is a strict increase when β > π/2 andψ > 0 because E⁰_(n+1) must account from the drop in potential energy from stepnto stepn+1. . . 47 3.7 Phase space, (φ, ˙φ), of the robot walking down a slope for 5 steps with no control

mechanism. We can see that the energy associated to the movement, from step to step, is converging since the differences become smaller. . . 48 3.8 Regions of parameter space associated to the existence of limit cycles. The colours

correspond toE/mlgand represent the energy necessary to maintain a limit cycle trajectory with a givenψandβ. Highlighted at red are the solutions in whichE/mlg=1, which is the energetic sufficient condition in order for the movement to take place. Due to the singularity associated to the term sin²β, the energyE/mlghas a big variation whenβis small. Whenβis big, the variations associated toE/mlgare small. Generally, whenβ<π/4 and, the lower isψ, the larger isE/mlg. . . 49

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3.9 Collision of the mass with a floor given by f₁(x) =_{0.4 cos}(_0.2+3x), an angle between the two legs at the transition instants of β = 0.525 rad and length associated to each leg ofl = 1 m. The green arrows near of the foot of the standing leg in steps 1 and 2 symbolize that the angle of attack is bigger than the exit angle in those steps, as required by Eq. (3.34). Given this, the robot does not fail steps 1 and 2. However, in step 3, the mass collides with the floor before a solution can be found in Eq. (3.4). . . 51 3.10 Robot walking the step 2 from the example in Fig. (3.9). The dashed robot represents

where step 3 would supposedly ends, whent=tf₍₃₎, if the robot hadn’t collided with the floor. Given that the step length is always the same, a circumference was drawn from the foot of the standing leg in step 3 to determine the fictional position of the foot of the standing leg in step 4. The red arrow symbolizes a contradiction of the requirement in Eq. (3.34) in the sense that in step 3, the exit angle must be smaller than the angle of attack, and in this case, it is not. The mass ends up colliding with the floor in step 3. . . 52 3.11 Example of the Failing Condition 1. In Fig. (3.11(a)), the foot of the imaginary trailing

leg at the start of the step is above the floor, as pointed by the green arrow, and as a consequence, it is possible to increment step 3. In Fig. (3.11(b)), on the other hand, the foot of the imaginary trailing leg is below the floor, as indicated by the red arrow, and therefore, the mass collides with the floor making it impossible to increment a step. . . . 53 3.12 Initial and final states of the robot in the limit which Failure Condition 1 provides a

successful iteration of the standing leg, with a floor f of type valley, by taking the plus sign of Eq. (3.35), forβ =π/3 andψ= π/6. In this case, the time elapsed in the step is null since we are in the critical situation in which the step can be incremented. Ifψ was higher, a step would not be possible and the mass would collide with the floor. . . 54 3.13 Internal collision of the standing leg with the floor. In this figure, the robot walks on a

surface with a local maximum. The trajectory of the mass is represented by a dashed segment and there are three important robot instances. At the ends of the trajectory, the initial and final instances of the step are represented. The third instance of the robot is when the standing leg crosses the tangent line of the floor that passes through the foot.

In this third instance, the standing leg is drawn red meaning that if its angle is lower than this minimum, which is represented byφ_Min, an internal collision occurs and the step fails. . . 55 3.14 Illustration of the Failure Condition 2 in a step. In both figures, the same floor was

used f(x) =sin(2.5(0.44+x))and the same leg lengthl = 1 m, however, the value of β is smaller for Fig. (3.14(a)), which justifies the internal collision of the standing leg in Fig. (3.14(b)). In both of the figures, three instances are portrayed, the initial instant of the step, which corresponds to the filled robot on the left, the other filled robot corresponds to the final state in this step, the dashed robot corresponds to the state when the standing leg has its leg at the minimum angle according to Eq. (3.37). . . 56

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3.15 Critical incrementation of the standing leg through a surface function Eq. (3.35), taking the minus sign, considering Failing Condition 2. In this figure, the critical value ofψ, the slope, depends ona, how far is the foot of the standing leg from the maximum, and β, the openness between the legs. The angleβin the final step was decomposed as the sum of angles the two adjacent rectangle triangles. The associated parameters areβ= 2π/3 rad,a=0.75 m,l=1 m andψwas obtained by Eq. (3.38), withψ=0.3029 rad. . 57 3.16 Critical incrementation of the standing leg regarding Failure Condition 2 by taking into

account thata=land selecting the minus sign of the surface function Eq. (3.35). With a= l, the aperture of the legs is minimized, thus, maximizing the number of distinct floors, with which the iteration of standing leg through a local maximum can succeed.

The problem is transformed from a bidimensional problem to a unidimensional problem, with the critical slope ψ only depending of β. The associated parameters are, ψ=3π/16 rad,β=π/4 rad. . . 58 3.17 Maximum angle that can be achieved by the slope of a surface functionf(x)as a func-

tion ofβto successfully increment a step through any type of continuous floor. When β < π/3, the robot is more sensible to valleys, thus the criteria associated with local minimum in Eq. (3.36) is more important than the criteria associated with local maximum. Whenβ>π/3, the more important criteria is reversed with the one associated with local maximum in Eq. (3.42) having more relevance against the criteria associated to local minimnum. Atβ=π/3 there is an intersection, meaning that there is no differ- ence, in this case, whether the floor has a more steep maximum, or minimum. At this intersection, the associated angle of the slope is maximized atψ=π/6, thus,β=π/3 maximizes the number of floors in which that can be successfully incremented. . . 59 3.18 Incrementation of 5 steps of the robot with β = 5π/9rad regarding the parameters

in the set of Eqs. (3.57) and (3.58) with the floor f of Eq. (3.56). All the steps were successfully incremented. . . 63 3.19 Incrementation of 5 steps of the robot with β = 2π/9rad regarding the parameters

in the set of Eqs. (3.57) and (3.59) with the floor f of Eq. (3.56). All the steps were successfully incremented. . . 64 3.20 Incrementation of 5 steps of the robot on top of the family of floors of the type in

Eq. (3.60) withr=30 and with the parameters in the set of Eqs. (3.61) and (3.62). With β= π/3 all the steps are successful even when the frequency of the floor approaches infinity. . . 65 3.21 Incrementation of 4 steps of the robot with parameters in the set of Eqs. (3.64) and (3.65),

along a floor with obstacles given in Eq. (3.63). The robot can pass first obstacle, the first step function, with an height of 0.105 m. In step 3, the robot tries to overcome a second step function of height 0.185 m, but since this value is above the minimum of the trajectory of the foot of the trailing leg, the foot intercepts the floor, rendering an invalid simulation. . . 67

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3.22 Differencey_TF(t)−f(x_TF(t))on the second, third and fourth steps of the simulation illustrated in Fig. (3.21). On step 2, this difference reaches 0 at the minimum of the trajectory, therefore, the height of the first step function of the floor corresponds to the maximum height allowed for the trailing leg to overcome this obstacle. On step 3, the height of the step function was increased and the differenceyTF(t)−f(xTF(t))reached negative values, thus failing the step because of the collision between the foot and the floor. . . 68 3.23 Incrementation of 4 steps of the robot with parameters in the set of Eqs. (3.64) and (3.66),

along a floor with obstacles given in Eq. (3.63).The robot can pass first obstacle, the first step function, with an height of 0.105 m. This time, in step 3, with ane>0, overcomes the second step function of height 0.185 m. . . 69 3.24 Differencey_TF(t)−f(x_TF(t))on the second, third and fourth steps of the simulation

illustrated in Fig. (3.23) withe > 0. On step 2, this difference is higher than 0 at the minimum of the trajectory, therefore, it is easier for the robot to passthe first obstacle comparatively to the previous case . On step 3, the height of the step function is increased to 0.185 m, and this time the differenceyTF(t)− f(xTF(t))reached 0. The increase on the parametereallowed for the robot to increased the maximum height that the robot can overcome in phaseb). . . 69 A.1 Difference between the solutions of the equation of motionφ(t), associated to the im-

plicit and the numerical method. The differences are of the order of 10⁻⁸ |φ(t)|, therefore there is no relevant difference between the numerical and the implicit method. A-3

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List of Tables

3.1 Quantities associated to the 5 steps of the robot in Fig. (3.18). In this example, the robot due to the high value ofβtravels a significant distance. This gaiting type is not good for saving energy since the projection of the velocity from the step before is not the same as the next step, therefore, the robot needs to stop the movement from the step before to begin another step with energyE. Due large values of T_(n), the frequencies associated to the hip and the knee are considerably small. . . 63 3.2 Quantities associated to the 5 steps of the robot in Fig. (3.19). In this example, the

openness between the legs is smaller and therefore, the time elapsed in a step is smaller and the distance traveled is also smaller. However, reaching step 3, the robot was almost reaching the limit cycle in the sense that the energy needed to add in order to conserve Ewas close to 0. Steps 4 and 5 were from the robot passing the minimum of f, and we can see that for a smaller value of β, the time elapsed in each step is much smaller. The reduced time of each step, for steps 4 and 5, is explained by Failing Condition 1 because the slope of the floor shortens the time for the next step to occur.

This effect increases the frequencies associated to the hip and the knee. . . 64 3.3 Quantities associated to the 5 steps of the robot in Fig. (3.20). For r very large, the

floor becomes a flat surface, thus, the easier it is to increment the trailing leg since the amplitude is smaller. This explains the fact that between steps there was a small variation in all quantities. . . 66 3.4 Quantities associated with the simulation of the robot in Fig. (3.21). This table is very

similar to the same results on a flat surface since there is no height variation from step to step associated to the standing leg. . . 68 3.5 Quantities associated with the simulation of the robot in Fig. (3.23). Similarly to the

previous case, this table is very similar to the results obtained on a flat surface with a difference that in this caseeis bigger than 0, and therefore,ωHc₍_n₎<0. . . 70

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List of Symbols

m Mass of the robot located at the hip . . . 12 l Length of the legs of the robot . . . 12 β Parameter of the standing leg that corresponds to the fixed angle of the

legs when the robot reaches a step transition . . . 12 α Inner angle measured at the foot of the isosceles triangle formed by the

mass and the two legs when the robot reaches a step transition . . . 12 g Gravitational acceleration . . . 13 φ Angle of the standing leg measured from a horizontal line passing through

the foot, to the mass . . . 13 φ˙ Angular velocity of the standing leg . . . 13 E Parameter of the standing leg that corresponds to the mechanical energy

of the robot . . . 13 t_i₍_n₎ Time then^thstep starts . . . 14 t_f₍_n₎ Time then^thstep ends . . . 14 φ0 Parameter of the standing leg that corresponds to the angle of the standing

leg on the first step,φ t_i₍₁₎

. . . 15 T₍₁₎ Duration of step 1 defined in Eq. (A.2) . . . 18 l1 Length of the thigh . . . 20 l2 Length of the calf . . . 20 λ Parameter of the trailing leg that corresponds to the ratio between the

length of the thigh and the length of the leg,l1/l . . . 20 θ Angle measured from the standing leg to the thigh of the trailing leg in

the intervalθ∈]−π,+π[ . . . 20 µ Angle measured from the thigh of the trailing leg to the calf in the interval

µ∈]0, 2π[ . . . 20 (xH(t),yH(t)) Position of the hip of the robot on the(x,y)plane at timet. . . 21 (xSF₍_n₎,ySF₍_n₎) Position of the foot of the standing leg associated to stepnon the (x,y)

plane . . . 21 (xTK(t),yTK(t)) Position of the knee of the trailing leg of the robot on the(x,y)plane at

timet . . . 21 (xTF(t),yTF(t)) Position of the foot of the trailing leg of the robot on the(x,y)plane at timet 21

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ta₍_n₎ Time associated to stepnwhen phasea)of the trailing leg starts . . . 22 t_b₍_n₎ Time associated to stepnwhen phaseb)of the trailing leg starts . . . 22 τ_b₍_n₎ Parameter of the trailing leg that corresponds to ^t^b⁽ⁿ⁾_T^−t^a⁽ⁿ⁾

(n) . . . 22 tc₍_n₎ Time associated to stepnwhen phasec)of the trailing leg starts . . . 22 τc₍_n₎ Parameter of the trailing leg that corresponds to ^t^c⁽ⁿ⁾_T^−t^a⁽ⁿ⁾

(n) . . . 22 e_(n) Parameter of the trailing leg that corresponds to an off set of the angleθ

on the final instant of phaseb) . . . 23 γ_(n) Paramater of the trailing leg that corresponds to the minimum ofµat the

start of phaseb)of the trailing leg . . . 23

~vf₍_n₎ Velocity of the mass of the robot at the end of stepn . . . 29

~vi₍_n₎ Velocity of the mass of the robot at the start of stepn . . . 29 v_fk₍_n₎ Projection of the velocity at the end of the previous step on the polar ver-

sor associated to the next step . . . 31

~v⁰_(n+1) Velocity added instantly on stepn+1 in order for the mechanical energy of the robot to remain the same . . . 31 E_(n+1)⁰ Energy added at the start of the stepn+1 in order to maintain the me-

chanical energy of the robot fixed . . . 35 σ_(n) Angle that the segment which connects the two feet makes with an hori-

zontal floor on the final instant of stepn . . . 44 ψ Angle the floor makes with a horizontal line . . . 44 ω_Ha₍_n₎ Angular frequency applied to the hip joint at the start of phasea). . . 62 ω_Hc₍_n₎ Angular frequency applied to the hip joint at the start of phasec). . . 62 ω_Ka₍_n₎ Angular frequency applied to the knee joint at the start of phasea) . . . 62 ω_Kb₍_n₎ Angular frequency applied to the knee joint at the start of phaseb) . . . 62

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1

Introduction

1.1 Motivation

One of the key factors that distinguish humans from other primates is their capability of having a bipedal walking/running gait. This aspect served as an evolutionary mark regarding our movement in the sense that we could access a much higher mobility and a bigger vision range than before against our preys and our predators. This along with the development of technology allowed for humans to have a bigger advantage against natural conditions and therefore contributed for our survival as species. Since the way we walk is unique and very complex [4] regarding the rest the animal kingdom, an effort has been made to reproduce the same type of walking and running in robots and other types of machines.

1.2 State Of The Art

In an attempt to reproduce the same effects of human walking and running two models were cre- ated with entirely different dynamics. Walking was developed [4–6] as an inverted pendulum model, while running was developed as a spring-mass model [1, 7]. With these models, several properties associated to walking or running could be obtained such as the kinetic and potential energies, the speed and frequency as well as other aspects for the associated gait. The spring-mass model was able to explain the basics of running. Regarding walking, it was verified that the inverted pendulum model led to an incomplete description [8]. With this in account, a two compliant leg model is pro- posed [2] consisting of two springs attached to a mass, being able to reproduce the same effects of walking with a relatively simple description despite the large complexity of the movement. In this paper, the stable parameter map is determined in relation with the variation of physical parameters such as the spring stiffnessk. Also, it is shown that walking and running can be encapsulated into the same model with the energy associated to the system being the mediator between walking and running. After the success, 5 types of walking gaits are determined using the same model, and, to test the robustness of the model the saddle and attraction region associated to the parameter space are determined [9].

Regarding more realistic spring models, an extensive analysis was done on several types of walking [10]. In this analysis, a review was made on biped walking, the motion of four legged creatures, namely, the traditional quadruped animals like dogs and cats as well as the motion of six legged organisms such as cockroaches for example. Besides studying and comparing the dynamical properties of these walking systems the neurological and mechanical aspects of locomotion are explored, where the muscles are modeled as activated springs regarding the respective neurological activation function. It is also shown that legged locomotion models are inherently stable with minimal neural feedback and, even though the neural and mechanical architecture has a potentially high number of dimensions associated to them, they effectively collapse into a much lower number through the ap- plication of phase response curves, averaging theory and the symmetry associated to these systems.

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1.2.1 Running/Hopping Model with Springs

In this model a spring-mass system is introduced by connecting the center of mass of a robot to a massless spring. This system describes the dependency of the different physical variables that charecterize running/hopping. Hopping forward can be described by a non-linear system of equations described by an inverted pendulum spring propagating forward in which there are two phases, the ground phase

¨

x=xω² l

px²+y²−1

!

, (1.1)

y¨=yω² l

px²+y² −1

!

−g, (1.2)

and the aerial phase, where

¨

x=_0, _(1.3)

¨

y=−g. (1.4)

In this system,mis the mass, ω² = _m^k is the natural frequency of the spring,g is the gravitational acceleration,xis the horizontal displacement,yis the vertical displacement andl = ^qx²₀+y²₀is the rest length of the spring.

Figure (1.1) helps to illustrate the model presented. In this figure, the spring touches the ground, starting the ground phase, with an angle of atackα, with an initial velocity~vwhere βis the angle that the velocity makes with the ground. After the spring touches the ground, it is compressed in the normal direction, and at the same time, the mass propagates forward since the spring rotates around its pivot. After the spring reaches the same height as when it first touched the ground, it starts the aerial phase where the spring becomes disconnected from the ground until the mass reaches an height of,l₀cosα, wherel₀is the rest length of the spring, restarting the ground phase.

Figure 1.1: Running/hopping model described in [1]. This model displays the basic dynamics of running/hopping. It is charecterized by two phases, the ground phase and the aerial phase. The angle which the leg returns to the ground is the same as in the previous step. In this scheme,~vis the velocity,mis the mass, αis the angle of attack andβis the angle the velocity makes with the ground.

The determination of the physical solutions are determined by narrowing the range of the parameters studied within the set of possible solutions. With these parameters, we need a small subset of them,

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the vector of landing velocity and the length of the leg to determine the state of the system and its evolution. Regarding running, a human subject maximizes the amount of energy that can be stored and delivered elastically [1]. This can be achieved by maximizing the contact length and this is why animals use flat angles associated to the landing velocity.

1.2.2 Walking Model with Springs

In this model [2], when one leg is on the ground, the system is equivalent to an inverted spring pendulum, this is called the single support phase. When the model is not in a single support phase, it is in a double support phase, where the two springs in this phase influence the movement of the center of mass at the same time reproducing a bipedal spring-mass walking. The springs accumulate and deliver energy so that the system remains convervative with no energy losses.

In the single support phase, the dynamics of the center of mass is as follows x¨= ^F¹

m x−x_t1

l1 , (1.5)

y¨= ^F¹ m

y−y_t1 l1

−g, (1.6)

where(x,y)are the coordinates of the center of mass and(x_ti,y_ti)are the coordenates of the respective toes for the leg 1 and leg 2. In the double support phase, the dynamics are similar to the previous case, with the difference that we are in the presence of two springs, this is,

¨ x= ^F¹

m x−xt1

l₁ +^F² m

x−xt2

l₂ , (1.7)

¨ y= ^F¹

m y−yt1

l₁ + ^F² m

y−yt2

l₂ −g, (1.8)

withFibeing the force applied on the mass by the respective leg,

Fi=k(l0−li)≥0 i=1, 2 , (1.9)

l0is the natural length of the spring,liis the respective length, l_i=

q

(x−x_ti)²+ (y−y_ti)² i=_{1, 2 .} _(1.10) Figure (1.2) illustrates this walking model. Single support to double support transitions occur when the vertical velocity of the center of mass is negative, and, the mass drops to a height of,ysinα, whereαis the angle of attack. Double support to single support transitions occur when the spring deflectionl0−l_iof one of the legs return to 0.

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Figure 1.2: Bipedal spring-mass walking model described in [2]. The initial velocity is given by~v, in this case, β = 0, this is the velocity is initially parallel to the ground,α_TD is the angle of attack for each leg, this is, the aperture that the leg has when it starts the double support phase.

This model is energetically conservative, therefore, the energyEis a constant with

E= ^k(l0−yn)²

2 +mgyn+mv²_n

2 . (1.11)

We can express the absolute value of the velocity in terms of the energy. In this model, the only parameters in the initial conditions that can alter the stability of the system for a certain angle of attack α, and energy E are β₀, the angle of the initial velocity with the ground and y0, the initial vertical position of the center of mass.

1.2.3 Inverted Pendulum Walking Model

In this thesis, one of the objectives is to develop, study and explain an algorithm capable of adapting to an irregular surface. The model that will serve as basis to our study will be the instant collision inverted pendulum model. The inverted pendulum model has differences from a real walking gait [11]. By ensuring that our model is not elastic, we reduce complexity of the problem regarding the study of irregular surfaces. This simplification is important since the limits of the inverted pendulum model were never explored when studying irregular surfaces, and also, because some of the more complex walking models can be seen as a degenerate case of the inverted pendulum model. For example, if we take the spring model in the limit that the spring stiffness is very high,k→_{∞, we will} return to the inverted pendulum model. We can then take information about the inverted pendulum model and apply it to more complex models. The inverted pendulum model used in [3, 12] is a minimal biped model which has a point mass located on the hip and two massless legs. Since it is a

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simple pendulum, the knees and feet included are massless, and the legs are incompressible. In this walking model, there are two phases associated, namely, a continuous phase, where the mass has the dynamics of a normal inverted pendulum, and the switch phase, which is associated to the leg switch and it is performed in an instantaneous way. During the continous phase, if we take into account Fig. (1.3) we have that,

θ¨=g/lsinθ, (1.12)

whereθis the stance angle andlis the length of the leg.

Figure 1.3:Continuous phase described in [3]

Figure (1.4) represents the switch phase. In this phase, both of the legs, the stance leg and the trailling leg apply an instantaneous impulse from the ground along the two legs. One of the impulses is associated to the heel in the leading leg,_H~ and the other one is the push off~_Papplied to the trailing leg. The combination of these two impulses result in an instantaneous change of velocity in the center of mass. This leads to the following relation according the Newton laws,

m_V~⁺−m_V~⁻ =~_P+~_H. _(1.13)

We can solve Eq. (1.13) with thexaxis aligned along the velocity of the next step~_V⁺in Fig. (1.4) and theyaxis, perpendicular and aligned with the stance leg in the next step. The vectors in Eq. (1.13) have the following form in this specific Cartesian frame,

V~⁺= (V⁺, 0), _V~⁻ = (V⁻cos(−2θ),V⁻sin(−2θ)), ~_P= (Psin 2θ,Pcos 2θ), _H~ = (0,H). (1.14) Both of the equations associated to Eq. (1.13) can be written as

(mV⁺−mV⁻cos(−2θ) =Psin 2θ+0,

0−mV⁻sin(−2θ) =Pcos 2θ+H, (1.15)

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which leads to the following relations between the initial velocity ˙θ⁻and the end velocity ˙θ⁺ (θ˙⁺ =θ^˙⁻cos 2θ+Psin 2θ/ml,

θ˙⁻sin 2θ=Pcos 2θ/ml+H/ml. (1.16)

After the switch phase, the model transitions to the continuous phase with θ⁺ = −θ⁻ where the pendulum will swing around its pivot beggining another step. If we know the initial, ˙θ⁻, and final, θ˙⁺, angular velocities of the mass in the switch phase, we can know the forces associated to the push and the heel strike. This cycle of transitions from continuous to switch phase continues until the robot falls or it is demanded to stop.

Figure 1.4:Switch phase described in [3]

One of the first robots to be implemented in the context of the inverted pendulum model was developed by McGeer, in which the concept of passive dynamic walking was introduced [13]. From the inverted pendulum model in Figs. (1.3) and (1.4), the heel strike applies an impulse which partially stops the motion from the previous step, however, the larger is the velocity from the step before, the bigger will be the velocity of the current step. If we have a descending floor with a fixed slope, the ending velocity when step ends is bigger than in the case of a flat floor. Therefore, an equilibrium is possible, and in this situation, the gravitational energy added, is compensated by the heel strike, thus providing a stable gait. Passive dynamic walking corresponds to this type of walking gait associated to a robot that doesn’t require an external battery or motor in order to keep walking in a stable manner.

With access to the Poincaré map associated to a robot going down a surface, i.e. by recording the velocity and the angle at the start of each step for various initial conditions, and by determining the eigenvalues, λ₁, λ₂, of the Jacobian matrix associated to the Poincaré section from step to step, the existence of a limit cycle can thus be proven if|λ₁| < 1 and|λ2| < 1 [14], which confirms that for a given descending floor with constant slope, there can be a walking stable gait powered by gravity.

Another passive dynamic walker with knees was also implemented [15]. In this work, the step of the robot was divided in two phases, one where the knees were locked and another where the knees were unlocked. Besides the walking model developed, the basin of attraction was determined, which

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is associated with the stable gaits of the passive dynamic walker going down a slope. An active dynamic walker [16] was also developed, meaning there is an active control in each step, in which mechanical energy of the inverted pendulum model was fixed for each step, allowing the robot, also in two phases, using the articulation of the knee, to successfully walk along a straight line on a flat surface.

1.3 Original Contributions

In this thesis, we introduce an active dynamic walking model with a mass located at the hip and two massless legs. The robot based in this model is developed with knees, capable of walking in irregular surfaces with a fixed angle between the legs and a fixed mechanical energy at the start of the step.

The robot can walk on irregular surfaces due to the fact that we control the time of the step given the dynamics of the mass, and, we have a specific geometric condition, presented in Eq. (3.4), which must be satisfied since the aperture between the two legs is fixed when the step starts. In this biped model, one of the legs evolves in time with the dynamics of motion of the mass of the robot, and the other is governed by a control actuation on a 2-link body which produces an articulated movement. This motion associated to the controlled leg is divided in 3 phases. We find that the energetic sufficient condition in order for the robot not to fall is that it’s energy must be superior tomlg, wheremis the mass of the robot, l the length of the legs, and g the gravity acceleration. In this model, the same result of the passive dynamic walker is also recovered when the mechanical energy is not actively fixed in the case of the robot going down a slope. Besides encountering the passive dynamic walker, the full descritpion associated to the passive dynamic walker between the energy, angle of the slope and aperture between the legs can be found in Eq. (3.30).

Regarding the solutions of the leg that supports the mass, we find that there are constraints regarding the type of floors which are possible to be incremented on. We find that there is a sufficient criteria in order to successfully increment a step without any collisions associated to this leg and the mass with the floor. If Sufficient Condition 3 is satisfiedwe can always guarantee that a step can be incremented for the family of surfaces f(x+x0), withx0∈Rregarding the supporting leg.

With the full set of parameters associated to the robot, we formulate the necessary conditions in order for the step to be possible for the articulated leg given an irregular floor which involves filtering simulations in which the interceptions in Eqs. (3.46) to (3.51) occur.

1.4 Thesis Outline

This thesis is divided in four chapters. The first chapter corresponds to the introduction, where a state of the art analysis regarding walking models is presented. The second part corresponds to the development an inverted pendulum model walking in a flat surface. The motion of the supporting leg and the articulated leg are explained in detail and also is explained how the robot conserves the mechanical energy in order to keep walking in a stable manner. The third part corresponds to walking in irregular surfaces, where it is explained how can the robot walk a step along a given surface f(x),

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also, how is the control mechanism modified for irregular surfaces, the sufficient conditions in order to increment a step and some miscellaneous examples to better illustrate the work of this thesis. The and fourth chapter corresponds to the conclusions, where some concluding remarks are done as well directions are provided in possible future works.

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2

Movement Along a Flat Horizontal Surface

2.1 Walking model

In this chapter we develop a walking model composed of a robot with a massmlocated in the hip, and two massless rigid legs of length l. These two legs have distinct functions. One of the legs supports the mass of the robot throughout the step and propagates this mass forward. This leg will be called the standing leg. The other leg, the trailing leg, resets the orientation of the standing leg from the previous step so that it has the proper initial conditions for the next step. This reset motion is composed of an articulated 2-link controlled movement via the hip joint that rotates the thigh and the knee joint that rotates the calf. This articulated motion allows for the robot to overcome obstacles.

We consider that a step begins when the standing leg connects with the floor, and a step ends when the standing leg leaves the floor. By transitioning to another step, the robot will instantly swap between standing and trailing legs in order to keep walking continually. An important characteristic about this model is that when a new step starts, the aperture between the standing and trailing legs is always the same. This means that the foot associated to the step before, the foot associated to the next step and the mass always create an isosceles triangle with the inner angle associated to the mass always being the same when there is a step transition. Figure (2.1) illustrates two important angles, αand β, that express the aperture between the legs between step transitions. The angleβwill be a parameter of the model which expresses how open the legs when a step transition occurs. The angle α which corresponds to the interior angle of the triangle centered in one of the foots which can be given by

α= ^π−β

2 . (2.1)

If(x_old_f,y_old_f)are the coordinates of the old foot associated to the standing leg at the final moment of a step, the new foot,(xnew_f,ynew_f), on a flat horizontal surface, has the following coordinates,

(xnew_f,ynew_f) = (xold_f,yold_f) + (2lcosα, 0). (2.2)

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α

β l

2 l cos (α)

( x

_old_f

, y

_old_f

) ( x

_new_f

, y

_new_f

)

m

Figure 2.1: Transition instant of the two legs between steps. The massm, the foot associated to the step before and the foot associated to the next step always create an isosceles triangle with the aperture between the legs corresponding to the angleβ. Although the angleβis the parameter of the model, both of the anglesαandβare equivalent since both express the degree of openness between the legs when a step transition occurs.

2.2 Standing Leg

The robot is transported by the inverted pendulum associated to the foot, the standing leg and the mass. The equation of motion associated to the mass of the robot is given by,

mφ¨+mg

l cosφ=0, (2.3)

where g is the gravity acceleration. The angle φis the angle of the standing leg measured from a horizontal line passing through the foot, to the mass. The pendulum is characterized in the phase space(φ, ˙φ)by the Hamiltonian function,

H= ¹

2ml²φ˙²+mglsinφ. (2.4)

We are interested in the domain where 0< φ<πand ˙φ ∈R⁰⁻since the propagation of the massm is done in the positive direction of thexxaxis. In the open interval ofφconsidered, Eq. (2.3) has an unique unstable fixed point at(φ, ˙φ) = (π/2, 0).

Since the inverted pendulum is an Hamiltonian system, the energyEis conserved for the step duration, which implies,

φ˙ =− s

2 E

ml² −^g l sinφ

, (2.5)

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where H = E, and the negative sign comes from direction of motion of the mass. IfE > mgl, the phase curves are defined for φ ∈ [0,π]. If 0 < E ≤ mgl, the phase curves are defined for φ ∈ [0, arcsin(E/mlg)]∪[π−arcsin(E/mlg),π]. We are not interested in the later situation since the pendulum can fall before the step ends, colliding with the floor, due to an energetic deficiency. If the pendulum has enough energy to achieve its vertical position,E >mgl, which is where the potential energy is maximum, there will be no constraints regarding the conclusion of the next step. Taking into account Eq. (2.5), the sufficient condition that assures the standing leg doesn’t fall in any kind of surface due to an energetic deficiency is

φ˙(t_i)≤ − r

2g

l (1−sinφ(t_i)), (2.6)

wheret_i corresponds to the initial time measured for the respective step. By determining the phase curves associated to the pendulum, a geometric criteria can be set that filters the solutions that re- spect the sufficient energy condition,E>mgl. Changing the energy, changes the phase curve of the pendulum, which changes the outcome of the fall. Figure (2.2) is a representation of the phase curves for the different energies.

0.5 1.0 1.5 2.0 2.5 3.0

ϕ

-

5

-

4

-

3

-

2

-

1 0

ϕ 

[ s ^-1 ]

E = 500 J E = 784 J E = 900 J

Figure 2.2:Phase curves for the inverted pendulum in Eq. (2.5) for,m=80 Kg,l =1 mg=9.8 m/s², and for various values of energyH=E, namely,E=_{500 J}<mlg(blue),E=_{784 J}=mlg(orange) andE=_{900 J}>mlg (green). The level of energy ofE=mlg=784 J, is the phase curve associated boundary of the energetic sufficient conditionE=mgl. Only solutions with ˙φ≤0 are taken.

One of the aspects of this basic model of locomotion consists in considering that only one leg, the standing leg, is in contact with the ground, and, when it starts to fall, it can be compensated by another leg with a different angle. Suppose the robot is on the step numbernin a set course. The initial instant associated to the stepn is designated byti₍_n₎, and the angle of the standing leg at this time, φ

t=ti₍_n₎

, is called the angle of attack associated to stepn. Likewise, the final instant associated to the step n is designated by t_f₍_n₎, and the angle of the standing leg at this time, φ

t=t_f₍_n₎ , is

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the exit angle associated to stepn. Three parameters can be associated to the standing leg, namely, φ₀=φ

t_i₍₁₎

, the angle of attack associated to the first step, which is also the initial condition of the system. The second parameter is the aperture angle between the former and the next leg measured at the time there is a transition of steps, β. The third parameter of the standing leg is the energyE associated with the pendular movement of the step in Eq. (2.4). On a flat surface, the angleαfrom Fig. (2.1) and Eq. (2.1) is the same as the exit angle associated to the first step since both measurement references coincide with

φ

tf₍₁₎

=α= ^π−β

2 . (2.7)

2.2.1 Solution of the Equation of Motion

There are essentially two ways of deriving a solution to Eq. (2.3), either numerically, or implicitly.

Regarding the implicit method, an expression relating the various quantities introduced that involves the Jacobi Amplitude function and its inverse, the Elliptic Integral of the First Kind, allows to express the behavior of the mass as a function of time. Regarding the numerical solution, we used a fourth order Runge-Kutta integration with theMathematica 12.0software. The precision and accuracy were set in order for the difference between the implicit and numerical solutions to be of the order of 10⁻⁸. Both the numeric and implicit method are described in detail in appendix A.Figure (2.3) is an illustration that corresponds to the evolution in time of the standing leg in the first step with the following parameters,

φ₀=1.767 rad,E=900 J,β=1 rad,l=1 m,m=80 Kg. (2.8)

Control of a Robotic Leg for Walking on Irregular Surfaces