Step Transition

If no action is performed when a step transition occurs, the mass of the pendulum falls to the floor.

We must have a transition state which marks the end of the step and the beginning of a new one.

Figure (2.4) and the respective list bellow, illustrate and carefully detail the walking step algorithm including the transition between steps.

i)t_i(1)

ϕ^ti₍₁₎=ϕ0

2 1

β

ii)t>t_i(1)

2

1 l2

l1

iii)tf₍₁₎

ϕ^tf₍₁₎=α

π-α

1 2

β

Figure 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(t_i(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.

i) Representation of the standing leg (leg 1) and the trailing leg (leg 2) at the start of step 1,t=t_i(1). The initial angle of the standing leg isφ

t_i(1)

= φ₀ and the initial moment of the first step occurs att = t_i(1). The angleφ₀cannot be higher thanπ−α, otherwise the trailing leg would cross the floor, rendering an invalid step. The angleφ0, cannot be lower thanα, otherwise it is not possible for the legs to have aperture angleβwhen a step transition occurs. On a flat horizontal surface the parameterφ0is restricted because of the characteristics of the floor. This limitation is expressed as

α<φ₀<π−α. (2.9)

Due to the degree of freedom associated to the parameterφ₀and the fixed aperture angle ofβ between steps, the foot associated to the trailing leg is not in contact with the floor when the first step starts in Fig. (2.4). By feeding into the system the parameterE, which corresponds to the energy of the pendular motion, the initial velocity of this leg is obtained in Eq. (2.5). The energy Eshould be higher thanmlgin order for the mass to cross its vertical position atφ=π/2.

ii) Representation of the standing leg (leg 1) and trailing leg (leg 2) after step 1 begins, at a time t >t_i(1). The articulated leg has 3 important points to refer to. The hip joint, where the mass is located, the knee and the foot. The hip joint and the knee have a distance ofl1, and the distance between the knee and the foot isl2. Together, the sum of these two lengths must be equal to the total length of the leg,l=l1+l2. At the same time the step begins and the standing leg starts its pendular movement, the trailing leg starts also its articulated motion. The function of the trailing leg is to reset the ending stance of the standing leg associated to the previous step, so that a new step can take place.¹.

iii) Representation of the standing leg (leg 1) and the trailing leg (leg 2) when step 1 ends, att=t_f(1). The angle between both legs when the step transition of steps occurs, isβ. The aperture between the legs at transition instants, can go from fully closed (β=0) to fully open (β=π),

0≤β≤π. (2.10)

The trailing leg, finishes the movement when the step ends, with the hip joint, knee and foot aligned making an angle of π−α with a horizontal line. In an instantaneous transition, the standing leg from step 1 becomes the trailing leg for step 2 and the trailing leg from step 1, becomes standing leg for step 2. Upon a step transition, the energyEassociated to the pendular motion of the mass is not lost from the previous step to the next.².

There are three consequences of this, firstly, the angles associated to the standing leg as well as the trailing leg have discrete instantaneous transitions. This way, the final time of step 1 can be defined as,

t_f(1) =t_i(1)+ lim

t→T₍⁻₁₎t, (2.11)

1In general, the articulated movement can be described by three phases and involve the simultaneous rotations of both the knee and the hip joint. The movement of the trailing leg is described more specifically in § (2.3).

2More information about the control mechanism that has to be implemented so that the energy associated to the motion of the mass is not lost can be found in § (2.4).

and the initial time of step 2 as,

t_i(2) =t_i(1)+ lim

t→T₍⁺₁₎t, (2.12)

whereT₍₁₎is the duration of step 1 and can be obtained from Eq. (A.2). Secondly, the angle of attack associated to the step 2 becomes,

φ t_i(2)

=π−α, (2.13)

since the trailing leg from the step before becomes the standing leg for the next step. Thirdly, the angular velocity of the mass on the second step can be obtained by Eq. (2.5) using the same value ofEfrom step 1, this is,

φ˙

ti₍₂₎

=− s

2 E

ml² −^g l sinφ

ti₍₂₎

. (2.14)

After the system finishes the step and transitions appropriately to another one, the robot begins another step by repeating the processes from items i) to iii). This cycle is repeated, until the robot is asked to stop.

Figure (2.5) is the representation of the standing leg and the mass in steps 1 and 2 according to the algorithm explained in items i) to iii) for a flat horizontal surface with the parametersφ₀ = 1.76756 rad,E=900 J andβ=1 rad.

ϕ

 = α ϕ (

)= π - α ϕ (

)= ϕ

ϕ

 = α

Step 1 Step 2

β

Figure 2.5: Standing leg and the trajectory of the mass in the first and second steps of the movement. 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).

From the situation in Fig. (2.5) it is possible to extract the information about the phase space in steps 1 and 2. Figure (2.6) is an illustration of the phase space of the both steps mentioned.

t

t

t

0.5 1.0 1.5 2.0 2.5 3.0

ϕ

-5 -4 -3 -2 -1 0

ϕ  [s

]

(a)Phase space associated to the step 1 of the standing leg and respective tran-sition to step 2 in Fig. (2.5). The event atti₍₁₎ corresponds to the standing leg in the starting position, namely, atφ(t_i(1)) = φ0. The standing leg, af-terwards has a pendular movement until t_f(1), when the step ends. The arrow symbols an instantaneous transition to the next step and the flow of the phase space. Since the surface is flat and horizontal and between steps the energyEof the mass conserved, the angular velocity remains the same as the angular velocity at the end of the step.

tf₍₂₎ ti₍₂₎

0.5 1.0 1.5 2.0 2.5 3.0 ϕ

-5 -4 -3 -2 -1

ϕ0

[s^-1]

(b)Phase space associated to the step 2 of the standing leg in Fig. (2.5). After the discrete transition, the standing leg starts the step 2 with its angle of attack ofφ(t_i(2)) =_π−α, has a pendular movement untilt_f(2) where the exit angle becomesφ(t_f(2)) = α. The exit angle at step 2 remains at α because the surface has not changed from step 1.

Figure 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 to E=mlg=784 J.