Passivity-Based Control Strategy for Humanoids

Abstract

This chapter covers passivity-based controls for humanoid robots. Passivity-based control has advantages over general active controls that require a large amount of energy consumption while the amount of energy that they carry is limited. First, passivity-based control is reviewed. Euler-Lagrange systems, examples of which are humanoid robots, and their passivity are described. Analysis on passive bipedal walkers is done based on the impact model for the contacts between the swing leg and the ground. This covers how a Poincaré map can be used in finding stable passive locomotion conditions. The concept of the passive bipedal walker is further expanded to an active control of a humanoid robot in order to give it more adaptability to its changing environment. A passivity-based control that has the adaptability of the ground slop and a simple passivity-based active control that increases the attraction basin of the limit-cycle of locomotion for more robustness are described.

Appendix

1.1 Notation

Let’s define some short notations for scalar and vector differentials with respect to a vector. Since some use different definitions, (Here, dp(q)/dq with $p(\cdot )\in \mathbb {R}$ and $q\in \mathbb {R}^n$ denotes a column vector, while some others may use it to denote rather a row vector.) it is necessary to clearly define the one used here.

For a function of q, h(q), where $q=\begin {bmatrix}q_1(t) & q_2(t) & \cdots & q_n(t)\end {bmatrix}^T\in \mathbb {R}^n$,

$$\displaystyle \begin{aligned} \dfrac{d h}{d q}\stackrel{\Delta}{=} \begin{bmatrix}\dfrac{\partial{h}}{\partial{q_1}} & \dfrac{\partial{h}}{\partial{q_2}}& \cdots & \dfrac{\partial{h}}{\partial{q_n}} \end{bmatrix}^T \in\mathbb{R}^n, \end{aligned} $$

(39)

is defined as a column vector and thus identical to the gradient of h w.r.t. q, ∇h.

Therefore, the derivative of h(q(t)) w.r.t. the time becomes

$$\displaystyle \begin{aligned} \dot{h}=\left( \dfrac{d h}{d q} \right)^T\dot{q}, \end{aligned} $$

(40)

which is simply a chain rule.

For a vector of functions, $f=\begin {bmatrix} f_1(q) & f_2(q)& \cdots & f_m(q)\end {bmatrix}^T\in \mathbb {R}^m$ where f _i (q)(1 ≤ i ≤ m) is a (scalar) function of $q\in \mathbb {R}^n$, let’s define the following:

$$\displaystyle \begin{aligned} \dfrac{d f}{d {q}}\stackrel{\Delta}{=} \begin{bmatrix} {\partial{f_1}}/{\partial{q_1}} & {\partial{f_1}}/{\partial{q_2}} & \cdots & {\partial{f_1}}/{\partial{q_n}}\\ {\partial{f_2}}/{\partial{q_1}} & {\partial{f_2}}/{\partial{q_2}} & \cdots & {\partial{f_2}}/{\partial{q_n}}\\ \vdots & \vdots & & \vdots \\ {\partial{f_m}}/{\partial{q_1}} & {\partial{f_m}}/{\partial{q_2}} & \cdots & {\partial{f_m}}/{\partial{q_n}} \end{bmatrix}\in\mathbb{R}^{m\times n}, \end{aligned} $$

(41)

which is called Jacobian. Thus,

$$\displaystyle \begin{aligned}\left( \dfrac{df}{dq}\right)_{ij}=\dfrac{\partial{f_i}}{\partial{q_j}}.\end{aligned}$$

1.2 Humanoid Dynamics

For Lagrangian $\mathcal {L}$ defined in Eq. (3),

$$\displaystyle \begin{aligned} \dfrac{\partial{\mathcal{L}}}{\partial{\dot{q}}}=M(q)\dot{q}, \end{aligned} $$

(42)

and thus

$$\displaystyle \begin{aligned} \dfrac{d}{dt}\left( \dfrac{\partial{\mathcal{L}}}{\partial{\dot{q}}} \right)=\dfrac{d}{dt}\left({M}(q)\right) \dot{q}+M(q)\ddot{q}. \end{aligned} $$

(43)

By the way,

$$\displaystyle \begin{aligned} \dfrac{d}{dt}\left({M}(q)\right)\dot{q}&= \left[\sum_{i=1}^n \dfrac{\partial}{\partial{q_i}}M(q) \dot{q}_i \right] \dot{q}\\ &=\sum_{i=1}^n \dfrac{\partial}{\partial{q_i}}M(q) \dot{q} \dot{q}_i \\ &=\begin{bmatrix} \dfrac{\partial{\left( M(q)\dot{q} \right)}}{\partial{q_1}} & \dfrac{\partial{\left( M(q)\dot{q} \right)}}{\partial{q_2}} & \cdots & \dfrac{\partial{\left( M(q)\dot{q} \right)}}{\partial{q_n}} \end{bmatrix} \begin{bmatrix}\dot{q}_1 \\ \dot{q}_2 \\ \vdots \\ \dot{q}_n\end{bmatrix}\\ &= \left( \dfrac{\partial{\left( M(q)\dot{q} \right)}}{\partial{q}} \right)^T \dot{q} {}. \end{aligned} $$

(44)

And,

$$\displaystyle \begin{aligned} \dfrac{\partial\mathcal{L}}{\partial{q}}=\dfrac{1}{2} \dfrac{\partial}{\partial{q}}\left[ \dot{q}^TM(q)\dot{q}\right]-\dfrac{\partial{V}}{\partial{q}}. \end{aligned} $$

(45)

Therefore, Lagrangian equation of motion , Eq. (2), becomes

$$\displaystyle \begin{aligned} M(q)\ddot{q}+\dfrac{d}{dt}\left( M(q)\right)\dot{q}-\dfrac{1}{2}\dfrac{\partial}{\partial{q}}\left[ \dot{q}^TM(q)\dot{q}\right]+\dfrac{\partial{V}}{\partial{q}}=\Xi_q \end{aligned} $$

(46)

Since Lagrangian equation of motion can be also expressed by

$$\displaystyle \begin{aligned} M(q)\ddot{q}+C(q,\dot{q})\dot{q}+g(q)=\Xi_q, \end{aligned} $$

(47)

$$\displaystyle \begin{aligned} C(q,\dot{q})\dot{q}&= \dfrac{d}{dt}\left( M(q)\right)\dot{q}-\dfrac{1}{2}\dfrac{\partial}{\partial{q}}\left[ \dot{q}^TM(q)\dot{q}\right]\\ &= \left( \dfrac{\partial{\left( M(q)\dot{q} \right)}}{\partial{q}} \right)^T \dot{q} -\dfrac{1}{2}\dfrac{\partial}{\partial{q}}\left[ \dot{q}^TM(q)\dot{q}\right], \end{aligned} $$

where Eq. (44) is used.