Study of compliance in human locomotion with computational
3.6 Dynamic reconstruction with optimal control
With the kinematic mapping, the joint angle trajectories could be recovered from motion cap- ture data, however they do not respect any dynamic constraint and we cannot gain dynamic information from these data. Therefore, a dynamic reconstruction (dynamic fit) as least square problem (LSQ) is carried out using the optimal control theory as described in Chapter2.2in combination with the 2D human model described in Section3.2. It consists in the mapping of the joint angle trajectories obtained from motion capture data to the rigid multi-body dynamic model taking into account all dynamic properties and constraints, fulfilling a certain specified objective function or set of objective functions.
The objective function in this case consists usually in the minimization of the error between the measurements and the model, but it also depends on the goal of the analysis. Also the definiton of states, controls, parameters and constraints are application oriented and will be defined in the next chapter for the studies of this thesis.
However, in any motion reconstruction optimal control problem, the constraints on the feet contacts during the different phases can be generalized.
Referring to the two cases we have shown in Sections3.5.1 and3.5.2, the equality and in- equality constraints are defined as follows.
Level ground and slope
In this case the number of phases is nph= 4, with two additional transition phases with zero time corresponding to the impacts.
• Phase 1 - Right foot flat
At the beginning of the phase, for t= s0, the foot has to be rigidly attached to the ground:
Rzt oe = 0 Rzheel = 0 ˙ Rzt oe = 0 ˙ Rzheel = 0 Lzt oe = 0 f(Rzt oe) ¾ 0 f(Rzheel) ¾ 0 (3.3)
where ˙Rand ˙L are the velocity of the indicated contact point of right or left foot, and f is a function that computes the ground reaction forces of a specified point.
During the phase, for t∈ (s0, s1): Lzt oe ¾ 0
Lzheel ¾ 0 f(Rzt oe) ¾ 0 f(Rzheel) ¾ 0
(3.4)
to ensure that the left foot does not go below the ground. • Phase 2 - Right toe At the beginning of the phase, for t= s1:
f(Rzheel) = 0 Lzt oe ¾ 0 Lzheel ¾ 0 ˙ Rzheel ¾ 0 f(Rzt oe) ¾ 0 (3.5)
to ensure the right toe lift off. During the phase, for t∈ (s1, s2):
Lzt oe ¾ 0 Lheelz ¾ 0 Rzheel ¾ 0 f(Rzt oe) ¾ 0
• Touchdown 1 - Right toe, left heel touchdown The time of transition phases is zero, therefore t= s2:
Lheelz = 0 Rzheel ¾ 0 Lzt oe ¾ 0 ˙Lz heel ¶ 0 f(Rzt oe) ¾ 0 (3.7)
where the velocity of the touchdown point changes.
• Phase 3 - Right toe, left heel During the phase, for t∈ (s2, s3):
Rzheel ¾ 0 Lzt oe ¾ 0 f(Rzt oe) ¾ 0 f(Lheelz ) ¾ 0
(3.8)
• Touchdown 2 - Right toe, left heel, left toe touchdown For time t= t3: Lzt oe = 0 Rzheel ¾ 0 ˙Lz t oe ¶ 0 f(Rzt oe) ¾ 0 f(Lheelz ) ¾ 0 (3.9)
• Phase 4 - Left foot flat
For the whole duration of the phase, for t∈ [s3, s4]: Rzheel ¾ 0 f(Rzt oe) ¾ 0 f(Lzt oe) ¾ 0 f(Lheelz ) ¾ 0 (3.10) Stairs up
In the case of stairs walking, additional environmental constraints as inequality constraints are added to ensure that the feet do not walk into the stairs. The number of phases is nph= 3, with one transition phase with zero time corresponding to the single impact.
• Phase 1 - Right foot flat, left toe
step: Rzt oe− zst ai r+ hst ai r = 0 Rzheel− zst ai r+ hst ai r = 0 Lzt oe− zst ai r = 0 ˙ Rzt oe = 0 ˙ Rheelx = 0 ˙ Rzheel = 0 ˙Lx t oe = 0 ˙Lz t oe = 0 Lheelz ¾ 0 f(Rzt oe) ¾ 0 f(Rzheel) ¾ 0 f(Lzt oe) ¾ 0 (3.11)
where zst ai ris the current height of the stair on which the left foot is stepping on, as this could be different from dataset to dataset, and hst ai r the height of the stairs step, which is fixed and known a priori from the environment setting.
During the phase, for t∈ (s0, s1):
f(Rzt oe) ¾ 0 f(Rzheel) ¾ 0 f(Lzt oe) ¾ 0
(3.12)
• Phase 2 - Right foot flat
At the beginning of the phase, for t= s1:
Rzt oe− zst ai r+ hst ai r = 0 Rzheel− zst ai r+ hst ai r = 0 ˙Lz t oe ¾ 0 f(Rzt oe) ¾ 0 f(Rzheel) ¾ 0 (3.13)
to ensure that right toe lifts off. During the phase, for t∈ (s1, s2):
Lt oex + xst ai r ¾ 0 Lzt oe− zst ai r ¾ 0
f(Rzt oe) ¾ 0 f(Rzheel) ¾ 0
(3.14)
where xst ai r is the position of the stair step in which the left foot should not enter, this is to ensure that environmental constraints are respected.
• Touchdown 1 - Right foot flat, Left foot touchdown For time t= s2: Lzt oe− zst ai r+ 2 ∗ hst ai r ¾ 0 Lzheel− zst ai r+ 2 ∗ hst ai r ¾ 0 ˙Lz t oe ¶ 0 ˙Lz heel ¶ 0 f(Rzt oe) ¾ 0 (3.15)
to ensure that the left foot is flatly positioned on the next step, therefore 2∗ hst ai r. • Phase 3, Right toe, left foot flat
For the whole duration of the phase t∈ [s2, s3]:
f(Rzt oe) ¾ 0 f(Lzt oe) ¾ 0 f(Lheelz ) ¾ 0
(3.16)
walking
The objective is to apply optimal control to the analysis of human gait from the dynamics point of view. The mapping of the motions is carried out as described in Section3.6. With this approach several aspects of human locomotion can be exploited, according to the formulation of the problem, e.g. in terms of how the states, controls, parameters, constraints and objective functions are being defined.
Here the feature of human walking being exploited is compliance, in terms of joint and bi- articular coupling stiffness. There is a large amount of literature on stiffness at joint level, which are mostly focused on level ground walking, with a much smaller amount of works on other walking scenarios. Some works focused on the analysis of kinematics and kinetics of slope walking[38,105] and stair climbing [13, 12], but there is a lack of studies focused on
joint stiffness.
In this chapter, our first goal is to undersand if and how compliance at joint and bi-articular level modulate during walking in different environments, then if and how this modulation influences the walking gait.
First the specific data used to carry out the analysis is being presented, then the two problems are shown in details with results and discussion.