Leg Centroidal Dynamics Based New Locomotion Principle of a Quadruped Robot with On-line Legged Motion Generation

Abstract

This paper proposes a novel legged locomotion principle for tracking the quadruped-robot base spatial motion which can generate automatically the trotting and walking gaits without flight phases. The principle is based on a new legged locomotion generator using the robot centroidal properties without a massless-leg assumption. Our quadruped robot is newly designed with symmetric legged dynamics properties using parallelogram mechanisms, and the corresponding generalized dynamics model is built. The centroidal momentum and dynamics models are established by combining the centroidal properties of the base and each leg, in which the virtual joint between the base and each leg CoM is used instead of the real joints. The stance legs follow the base motion with respect to contact constraints. The swing legs compensate for the delay influences (to centroidal motion) by the stance legs. By combining the whole-body kinematics model and the kinematics constraints relating to the contact constraints and the parallelogram mechanism, the adaptive swing-leg motion is achieved in the running process without pre-defined trajectories. When two swing legs land at different timings, the walking gait is applied. We also propose a new state machine for switching the locomotion gaits between walking and trotting, which also serves the torque controller which determines the dimension of the legged control states and the contact Jacobian. We apply our hierarchical torque controller to enable the robot with compliant dynamics properties. Therefore, our new locomotion principle integrates the system design, motion generation and whole-body torque control together, so that this one common framework is able to handle several locomotion gaits. To verify the usefulness and validity of our new locomotion principle based legged motion generator, we run several simulations: walking, trotting, turning, recovering from big lateral push force acting on the base.

Appendices

Appendix: A

Extension	Media type	Description
1	Video	A simulation video from
		Section 5.

Appendix: B

The scaling matrix λ_i in (20) is derived in the following equation by using the frame setting in Fig. 2,

$$ \boldsymbol{\lambda}_{i} = \left[\begin{array}{lll} \lambda_{z} & 0 & 0\\ 0 & \lambda_{x}\cos^{2}\theta_{i}+\lambda_{y}\sin^{2}\theta_{i} & (\lambda_{x}- \lambda_{y})\cos\theta_{i}\sin\theta_{i} \\ 0 & (\lambda_{x}- \lambda_{y})\cos\theta_{i}\sin\theta_{i} & \lambda_{x}\cos^{2}\theta_{i}+\lambda_{y}\sin^{2}\theta_{i} \end{array},\right] $$

(94)

where we use 𝜃_i to denote the leg i HAA joint angle $\boldsymbol {q}_{HAA_{i}}$ to save space. (λ_x, λ_y, λ_z) are constant scalars.

Appendix: C

The angular inertia at the robot CoM $\bar {\textbf {I}}_{G}$ is derived as below,

$$ \begin{array}{@{}rcl@{}} \bar{\textbf{I}}_{G} &=& {~}^{b}\boldsymbol{R}^{T}_{G} \bar{\textbf{I}}_{b}{~}^{b}\boldsymbol{R}_{G} -{~}^{b}\boldsymbol{R}^{T}_{G} \tilde{\boldsymbol{p}}_{G}\times m_{b}\tilde{\boldsymbol{p}}_{G}\times {~}^{b}\boldsymbol{R}_{G} + \\ &&\sum\limits_{i=1}^{4} \left( {~}^{b}\boldsymbol{R}^{T}_{G} \bar{\textbf{I}}_{i}{~}^{b}\boldsymbol{R}_{G}- {~}^{b}\boldsymbol{R}^{T}_{G} {~}^{i}\boldsymbol{p}_{G}\times m {~}^{i}\boldsymbol{p}_{G}\times {~}^{b}\boldsymbol{R}_{G} \right) , \end{array} $$

(95)

therefore, we can derive the following relationship by substituting (30b) and (37) into (95),

$$ {~}^{b}\boldsymbol{R}_{G}\bar{\textbf{I}}_{G}{~}^{b}\boldsymbol{R}^{T}_{G}\boldsymbol{\omega}_{b} = \bar{\textbf{I}}_{b}\boldsymbol{\omega}_{b} + \sum\limits_{i=1}^{4} \left[ \bar{\textbf{I}}_{i} \boldsymbol{\omega}_{b} - {~}^{i}\boldsymbol{p}_{G} \times m \boldsymbol{\omega}_{b} \times \tilde{\boldsymbol{p}}_{i} \right]. $$

(96)

Appendix: D

The variables in (69a) are defined as follows,

$$ \begin{array}{@{}rcl@{}} &\begin{aligned} \boldsymbol{B}_{sw_{f}} &=& \left[\begin{array}{lll} \frac{l_{r}}{2} & \frac{w_{r}}{2} & h_{r} \end{array}\right]^{T}, \quad \ \ \ \text{for FL leg}, \\ &=& \left[\begin{array}{lll} \frac{l_{r}}{2} & -\frac{w_{r}}{2} & h_{r} \end{array}\right]^{T}, \quad \text{for FR leg}, \end{aligned} \end{array} $$

(97a)

$$ \begin{array}{@{}rcl@{}} &\begin{aligned} \boldsymbol{B}_{sw_{h}} &=& \left[\begin{array}{lll} -\frac{l_{r}}{2} & \frac{w_{r}}{2} & h_{r} \end{array}\right]^{T} , \quad \text{for HL leg}, \\ &=&- \left[\begin{array}{lll} \frac{l_{r}}{2} & \frac{w_{r}}{2} & h_{r} \end{array}\right]^{T}, \ \ \ \text{for HR leg}, \end{aligned} \end{array} $$

(97b)

where h_r denotes the relative position between the base frame origin and each HAA joint origin, in z_b direction. l_r and w_r are referred in Fig. 2.

Appendix: E

The variables in (83) are defined as below,

$$ \begin{array}{@{}rcl@{}} \boldsymbol{J}_{f}&=&\left[\begin{array}{l} \boldsymbol{S}_{x}(\boldsymbol{J}_{p_{sw_{f}}} + \boldsymbol{J}_{p_{st_{h}}})\\ \boldsymbol{S}_{y}(\boldsymbol{J}_{p_{sw_{f}}} + \boldsymbol{J}_{p_{st_{f}}}) \\ \boldsymbol{S}_{z}(\boldsymbol{J}_{p_{sw_{f}}} - \boldsymbol{J}_{p_{sw_{h}}}) \end{array}\right] , \end{array} $$

(98a)

$$ \begin{array}{@{}rcl@{}} \boldsymbol{J}_{h}&=& \left[\begin{array}{l} \boldsymbol{S}_{x}(\boldsymbol{J}_{p_{sw_{h}}} + \boldsymbol{J}_{p_{st_{f}}})\\ \boldsymbol{S}_{y}(\boldsymbol{J}_{p_{sw_{h}}} + \boldsymbol{J}_{p_{st_{h}}}) \\ \boldsymbol{S}_{z}(\boldsymbol{J}_{p_{sw_{h}}} - \boldsymbol{J}_{p_{sw_{f}}}) \end{array}\right] , \end{array} $$

(98b)

$$ \begin{array}{@{}rcl@{}} \boldsymbol{e}_{f} &=&\left[\begin{array}{l} -\boldsymbol{S}_{x}(\tilde{\boldsymbol{p}}_{sw_{f}} - \tilde{\boldsymbol{P}}^{d}_{sw_{f}}) - \boldsymbol{S}_{x}(\tilde{\boldsymbol{p}}_{st_{h}} - \tilde{\boldsymbol{P}}^{d}_{st_{h}})\\ -\boldsymbol{S}_{y}(\tilde{\boldsymbol{p}}_{sw_{f}} - \tilde{\boldsymbol{P}}^{d}_{sw_{f}}) - \boldsymbol{S}_{y}(\tilde{\boldsymbol{p}}_{st_{f}} - \tilde{\boldsymbol{P}}^{d}_{st_{f}})\\ -\boldsymbol{S}_{z}(\tilde{\boldsymbol{p}}_{sw_{f}} + \tilde{\boldsymbol{p}}^{d}_{sw_{h}}) \end{array}\right], \end{array} $$

(98c)

$$ \begin{array}{@{}rcl@{}} \boldsymbol{e}_{h} &=&\left[\begin{array}{l} -\boldsymbol{S}_{x}(\tilde{\boldsymbol{p}}_{sw_{h}} - \tilde{\boldsymbol{P}}^{d}_{sw_{h}}) - \boldsymbol{S}_{x}(\tilde{\boldsymbol{p}}_{st_{f}} - \tilde{\boldsymbol{P}}^{d}_{st_{f}})\\ -\boldsymbol{S}_{y}(\tilde{\boldsymbol{p}}_{sw_{h}} - \tilde{\boldsymbol{P}}^{d}_{sw_{h}}) - \boldsymbol{S}_{y}(\tilde{\boldsymbol{p}}_{st_{h}} - \tilde{\boldsymbol{P}}^{d}_{st_{h}})\\ -\boldsymbol{S}_{z}(\tilde{\boldsymbol{p}}_{sw_{h}} + \tilde{\boldsymbol{p}}^{d}_{sw_{f}}) \end{array}\right], \end{array} $$

(98d)

where sw_f, sw_h, st_f and st_h represent the swing-front, swing-hind, stance-front and stance-hind leg index, and $\boldsymbol {J}_{p_{i}}$ is defined in (81), $\tilde {\boldsymbol {P}}^{d}_{i}$ is defined in (82).

Appendix: F

The dynamics properties for the centroidal-motion tracking task are achieved using (86) as follows.

$$ \begin{array}{@{}rcl@{}} \boldsymbol{{\varLambda}}_{G|c|m}&=&(\textbf{\textit{J}}_{G|c|m}\textbf{\textit{M}}^{-1}\textbf{\textit{J}}^{T}_{G|c|m})^{-1}, \end{array} $$

(99a)

$$ \begin{array}{@{}rcl@{}} \bar{\boldsymbol{J}}^{T}_{G|c|m}&=&\boldsymbol{{\varLambda}}_{G|c|m}\textbf{\textit{J}}_{G|c|m}\textbf{\textit{M}}^{-1}, \end{array} $$

(99b)

$$ \begin{array}{@{}rcl@{}} \boldsymbol{\mu}_{G|c|m}&=&\boldsymbol{{\varLambda}}_{G|c|m}\boldsymbol{J}_{G}\boldsymbol{M}^{-1}\boldsymbol{N}^{T}_{c|m}\textbf{\textit{C}} +\\ &&\boldsymbol{{\varLambda}}_{G|c|m}\left( \boldsymbol{J}_{G}\bar{\boldsymbol{J}}_{c|m}\dot{\boldsymbol{J}}_{c}\dot{\boldsymbol{q}}-\dot{\boldsymbol{J}}_{G}\dot{\boldsymbol{q}}\right), \end{array} $$

(99c)

$$ \begin{array}{@{}rcl@{}} \boldsymbol{\rho}_{G|c|m}&=&\boldsymbol{{\varLambda}}_{G|c|m}\boldsymbol{J}_{G}\boldsymbol{M}^{-1}\boldsymbol{N}^{T}_{c|m}\textbf{\textit{G}}, \end{array} $$

(99d)

where it is noticed that the subscript G can be replaced by b if tracking the floating-base motion is the first task. Similarly, the dynamic properties for the swing-leg motion tracking task are derived as below,

$$ \begin{array}{@{}rcl@{}} \boldsymbol{{\varLambda}}_{sw|G|c|m}&=&(\textbf{\textit{J}}_{sw|G|c|m}\textbf{\textit{M}}^{-1}\textbf{\textit{J}}^{T}_{sw|G|c|m})^{-1}, \end{array} $$

(100a)

$$ \begin{array}{@{}rcl@{}} \bar{\boldsymbol{J}}^{T}_{sw|G|c|m}&=&\boldsymbol{{\varLambda}}_{sw|G|c|m}\textbf{\textit{J}}_{sw|G|c|m}\textbf{\textit{M}}^{-1}, \end{array} $$

(100b)

$$ \begin{array}{@{}rcl@{}} \boldsymbol{\mu}_{sw|G|c|m}&=&\boldsymbol{{\varLambda}}_{sw|G|c|m}\boldsymbol{J}_{sw}\boldsymbol{M}^{-1}\boldsymbol{N}^{T}_{c|m}\textbf{\textit{C}}+ \end{array} $$

$$ \begin{array}{@{}rcl@{}} &&\boldsymbol{{\varLambda}}_{sw|G|c|m}\left( \boldsymbol{J}_{sw}\bar{\boldsymbol{J}}_{c|m}\dot{\boldsymbol{J}}_{c}\dot{\boldsymbol{q}}-\dot{\boldsymbol{J}}_{sw}\dot{\boldsymbol{q}}\right), \end{array} $$

(100c)

$$ \begin{array}{@{}rcl@{}} \boldsymbol{\rho}_{sw|G|c|m}&=&\boldsymbol{{\varLambda}}_{sw|G|c}\boldsymbol{J}_{w}\boldsymbol{M}^{-1}\boldsymbol{N}^{T}_{c|m}\textbf{\textit{G}}. \end{array} $$

(100d)

The general matrices $\boldsymbol {A}_{x_{i}}$ and $\boldsymbol {B}_{x_{i}}$ are derived as follows,

$$ \begin{array}{@{}rcl@{}} \boldsymbol{A}_{x_{i}} &=& \textbf{\textit{S}}\bar{\boldsymbol{J}}_{i|pre(i)}\boldsymbol{{\varLambda}}^{-1}_{i|pre(i)}, \end{array} $$

(101a)

$$ \begin{array}{@{}rcl@{}} \boldsymbol{B}_{x_{i}} &= &\boldsymbol{J}_{i|pre(i)}\boldsymbol{M}^{-1}\left( -\boldsymbol{\Gamma} + \sum\limits_{t=1}^{i-1}\boldsymbol{S}^{T}\boldsymbol{\tau}_{t}\right) - \\ &&\boldsymbol{{\varLambda}}^{-1}_{i|pre(i)} \left( \boldsymbol{\mu}_{i|pre(i)} + \boldsymbol{\rho}_{i|pre(i)}+ \boldsymbol{T}_{i|pre(i)} \right) , \end{array} $$

(101b)

where T_i|pre(i) combines all operational space forces of the task levels higher than i, as follows,

$$ \boldsymbol{T}_{i|pre(i)} = \boldsymbol{{\varLambda}}_{i|pre(i)}\boldsymbol{J}_{i}\boldsymbol{M}^{-1}\sum\limits_{t=1}^{i-1} \boldsymbol{J}^{T}_{t|pre(t)}\boldsymbol{F}_{t|pre(t)}, $$

(102)

where T_1|pre(1) = 0, and F_t|pre(t) is the operational space force for task i and it can be achieved by multiplying $\bar {\boldsymbol {J}}^{T}_{t|pre(t)}$ at the right side of the dynamics model in (87) as below,

$$ \boldsymbol{F}_{t|pre(t)} = \bar{\boldsymbol{J}}^{T}_{t|pre(t)}\left( - \boldsymbol{\Gamma} + \sum\limits_{j=1}^{t}\boldsymbol{S}^{T}\boldsymbol{\tau}_{j}\right), $$

(103)

where we use property $\bar {\boldsymbol {J}}^{T}_{t|pre(t)}\boldsymbol {N}_{pre(j)} = \boldsymbol {0}$ for j > t. In this way, T_i|pre(i) and F_t|pre(t) both depend on the prioritized impedance controller.