﻿ 类圆规双足被动行走模型及其稳定性
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 同济大学学报(自然科学版)  2017, Vol. 45 Issue (8): 1209-1217.  DOI: 10.11908/j.issn.0253-374x.2017.08.016 0

### 引用本文

AN Kang, LIU Chengju. Compass-Like Passive Dynamic Walking Model and Its Stability[J]. Journal of Tongji University (Natural Science), 2017, 45(8): 1209-1217. DOI: 10.11908/j.issn.0253-374x.2017.08.016.

### 文章历史

1. 同济大学 电机与信息工程学院，上海 201804;
2. 上海师范大学 信息与机电工程学院，上海 200234

Compass-Like Passive Dynamic Walking Model and Its Stability
AN Kang1,2, LIU Chengju1
1. School of Electronics and Information Engineering, Tongji University, Shanghai 201804, China;
2. College of Information, Mechanical and Electronic Engineering, Shanghai Normal University, Shanghai 200234, China
Abstract: This paper investigates a compass-like passive dynamic walking model. Horizontal and vertical DOFs (degrees of freedom) are designed at the stance foot to calculate the normal and friction forces, and describe the push-off and heel-strike processes. The passive dynamic walking gaits are calculated, and the local and global stabilities are analyzed. The results show that the walking gait is stable when the maximal absolute eigenvalue of the Jacobian is within the unit circle and the initial conditions of the walking gait is within the basin of attraction. These studies can not only provide guidance for understanding the motion mechanism of biped walking, but also help us for designing stable and efficient biped robot. For example, mechanical parameters improvement and efficient control strategy design.
Key words: stability    compass-like model    passive dynamic walking    biped locomotion    robotics

1 双足机器人模型 1.1 模型分析

 图 1 类圆规双足机器人模型的被动动态行走步态流程 Fig.1 A typical step of the compass-like passive dynamic walking model

1.2 摆动阶段的动力学分析

 $\left\{ \begin{array}{l} \frac{{\rm{d}}}{{{\rm{d}}t}}\left( {\frac{{\partial L}}{{\partial \dot q}}} \right) - \frac{{\partial L}}{{\partial q}} = 0\\ L = T - V \end{array} \right.$ (1)

 $\mathit{\boldsymbol{X}} = \left[ {\begin{array}{*{20}{c}} {{x_{{\rm{st}}1}}}\\ {{y_{{\rm{st1}}}}}\\ {{x_{{\rm{hip}}}}}\\ {{y_{{\rm{hip}}}}}\\ {{x_{{\rm{sw1}}}}}\\ {{y_{{\rm{sw1}}}}} \end{array}} \right]{\rm{ = }}\left[ {\begin{array}{*{20}{c}} {{x_0} - {l_{\rm{s}}}\sin \left( {{q_1}} \right)}\\ {{y_0} + {l_{\rm{s}}}\cos \left( {{q_1}} \right)}\\ {{x_0} - \left( {{l_{\rm{s}}} + {l_{\rm{t}}}} \right)\sin \left( {{q_1}} \right)}\\ {{y_0} + \left( {{l_{\rm{s}}} + {l_{\rm{t}}}} \right)\cos \left( {{q_1}} \right)}\\ {{x_0} - \left( {{l_{\rm{s}}} + {l_{\rm{t}}}} \right)\sin \left( {{q_1}} \right) + {l_{\rm{t}}}\sin \left( {{q_1} + {q_2}} \right)}\\ {{y_0} + \left( {{l_{\rm{s}}} + {l_{\rm{t}}}} \right)\cos \left( {{q_1}} \right) + {l_{\rm{t}}}\cos \left( {{q_1} + {q_2}} \right)} \end{array}} \right]$ (2)

 $\mathit{\boldsymbol{M}}\left( \mathit{\boldsymbol{q}} \right)\mathit{\boldsymbol{\ddot q}} + \mathit{\boldsymbol{C}}\left( {\mathit{\boldsymbol{q}},\mathit{\boldsymbol{\dot q}}} \right)\mathit{\boldsymbol{\dot q}} + \mathit{\boldsymbol{G}}\left( {\mathit{\boldsymbol{q}},\mathit{\boldsymbol{\gamma }}} \right) = \mathit{\boldsymbol{J}}_r^{\rm{T}}\mathit{\boldsymbol{u}}$ (3)

1.3 行走过程中的支撑足约束

 ${\mathit{\boldsymbol{J}}_{\rm{r}}}\left[ {\mathit{\boldsymbol{q}}\;\;\;\mathit{\boldsymbol{\dot q}}\;\;\;\mathit{\boldsymbol{\ddot q}}} \right] = 0$ (4)
 ${\mathit{\boldsymbol{J}}_\mathit{r}} = \left[ {\begin{array}{*{20}{c}} 0&0&1&0\\ 0&0&0&1 \end{array}} \right]$ (5)

 ${\mathit{\boldsymbol{J}}_\mathit{r}}\mathit{\boldsymbol{\ddot q}} = - {\mathit{\boldsymbol{J}}_\mathit{r}}\mathit{\boldsymbol{M}}{\left( \mathit{\boldsymbol{q}} \right)^{ - 1}}\left( {\mathit{\boldsymbol{C}}\left( {\mathit{\boldsymbol{q}},\mathit{\boldsymbol{\dot q}}} \right)\mathit{\boldsymbol{\dot q}} + \mathit{\boldsymbol{G}}\left( {\mathit{\boldsymbol{q}},\mathit{\boldsymbol{\gamma }}} \right) - \mathit{\boldsymbol{J}}_r^{\rm{T}}\mathit{\boldsymbol{u}}} \right)$ (6)

 $\mathit{\boldsymbol{u}} = {\left( {{\mathit{\boldsymbol{J}}_\mathit{r}}\mathit{\boldsymbol{M}}{{\left( \mathit{\boldsymbol{q}} \right)}^{ - 1}}\mathit{\boldsymbol{J}}_r^{\rm{T}}} \right)^{ - 1}}{\mathit{\boldsymbol{J}}_\mathit{r}}\mathit{\boldsymbol{M}}{\left( \mathit{\boldsymbol{q}} \right)^{ - 1}}\left( {\mathit{\boldsymbol{C}}\left( {\mathit{\boldsymbol{q}},\mathit{\boldsymbol{\dot q}}} \right)\mathit{\boldsymbol{\dot q}} + \mathit{\boldsymbol{G}}\left( {\mathit{\boldsymbol{q}},\mathit{\boldsymbol{\gamma }}} \right)} \right)$ (7)

1.4 足地碰撞过程

 ${y_0}/l + \cos \left( {{q_1}} \right) - \cos \left( {{q_1} + {q_2}} \right) = 0$

 $\left[ {\begin{array}{*{20}{c}} {q_1^ + }\\ {q_2^ + }\\ {x_0^ + }\\ {y_0^ + } \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {q_1^ - + q_2^ - }\\ { - q_2^ - }\\ {x_0^ - + l\left( {\sin \left( {q_1^ - + q_2^ - } \right) - \sin \left( {q_1^ - } \right)} \right)}\\ 0 \end{array}} \right]$ (8)

 $\left[ {\begin{array}{*{20}{c}} {\dot q_1^*}\\ {\dot q_2^*}\\ {\dot x_0^*}\\ {\dot y_0^*} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\dot q_1^ - + \dot q_2^ - }\\ { - \dot q_2^ - }\\ {\dot x_0^ - + l\left( {\dot q_1^ - + \dot q_2^ - } \right)\cos \left( {q_1^ - + q_2^ - } \right) - l\dot q_1^ - \cos q_1^ - }\\ {\dot y_0^ - + l\left( {\dot q_1^ - + \dot q_2^ - } \right)\sin \left( {q_1^ - + q_2^ - } \right) - l\dot q_1^ - \sin q_1^ - } \end{array}} \right]$ (9)

 $\mathit{\boldsymbol{M}}\left( {{\mathit{\boldsymbol{q}}^ + }} \right){{\mathit{\boldsymbol{\dot q}}}^ + } - \mathit{\boldsymbol{M}}\left( {{\mathit{\boldsymbol{q}}^*}} \right){{\mathit{\boldsymbol{\dot q}}}^*} = \mathit{\boldsymbol{J}}_i^{\rm{T}}\mathit{\boldsymbol{P}}$ (10)

 ${\mathit{\boldsymbol{J}}_i}{{\mathit{\boldsymbol{\dot q}}}^ + } = 0$ (11)

 ${\mathit{\boldsymbol{J}}_i}{{\mathit{\boldsymbol{\dot q}}}^*} + {\mathit{\boldsymbol{J}}_i}\mathit{\boldsymbol{M}}{\left( {{\mathit{\boldsymbol{q}}^*}} \right)^{ - 1}}\mathit{\boldsymbol{J}}_i^{\rm{T}}\mathit{\boldsymbol{P}} = 0$ (12)

 $\mathit{\boldsymbol{P}} = - {\left( {{\mathit{\boldsymbol{J}}_i}\mathit{\boldsymbol{M}}{{\left( {{\mathit{\boldsymbol{q}}^*}} \right)}^{ - 1}}\mathit{\boldsymbol{J}}_i^{\rm{T}}} \right)^{ - 1}}{\mathit{\boldsymbol{J}}_i}{{\mathit{\boldsymbol{\dot q}}}^*}$ (13)

 ${{\mathit{\boldsymbol{\dot q}}}^ + } = \left( {I - \mathit{\boldsymbol{M}}{{\left( {{\mathit{\boldsymbol{q}}^*}} \right)}^{ - 1}}\mathit{\boldsymbol{J}}_i^{\rm{T}}{{\left( {{\mathit{\boldsymbol{J}}_i}\mathit{\boldsymbol{M}}{{\left( {{\mathit{\boldsymbol{q}}^*}} \right)}^{ - 1}}\mathit{\boldsymbol{J}}_i^{\rm{T}}} \right)}^{ - 1}}{\mathit{\boldsymbol{J}}_i}} \right){{\mathit{\boldsymbol{\dot q}}}^*}$ (14)

2 周期的被动动态行走步态

 ${\mathit{\boldsymbol{q}}_{n + 1}} = f\left( {{\mathit{\boldsymbol{q}}_n}} \right)$ (15)

 ${\mathit{\boldsymbol{q}}_{n + 1}} = {\mathit{\boldsymbol{q}}_n}$ (16)

 ${\mathit{\boldsymbol{q}}_{\rm{c}}}{\rm{ = }}f\left( {{\mathit{\boldsymbol{q}}_{\rm{c}}}} \right)$ (17)
 图 2 机器人行走步态的庞加莱截面示意图 Fig.2 Poincare section of the walking gait

 图 3 机器人被动动态行走周期步态过程 Fig.3 Process of the periodic passive dynamic walking gait

 $\Delta V = mg{l_{{\rm{step}}}}\sin \gamma$ (18)
 图 4 机器人周期步态行走过程中的能量变化情况 Fig.4 Energy variation during the periodic walking motion

3 被动动态行走步态的稳定性

3.1 局部稳定性

 $\begin{array}{l} {\mathit{\boldsymbol{q}}_{\rm{c}}} + \Delta {\mathit{\boldsymbol{q}}_{n + 1}} = {\mathit{\boldsymbol{q}}_{n + 1}} = f\left( {{\mathit{\boldsymbol{q}}_n}} \right) = f\left( {{\mathit{\boldsymbol{q}}_{\mathit{c + }}}\Delta {\mathit{\boldsymbol{q}}_n}} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; \approx f\left( {{\mathit{\boldsymbol{q}}_{\rm{c}}}} \right) + \mathit{\boldsymbol{J}}\Delta {\mathit{\boldsymbol{q}}_n}{\rm{ = }}\mathit{\boldsymbol{qJ}}\Delta {\mathit{\boldsymbol{q}}_n} \end{array}$ (19)

 $\Delta {\mathit{\boldsymbol{q}}_{n + 1}}{\rm{ = }}\mathit{\boldsymbol{J}}\Delta {\mathit{\boldsymbol{q}}_n}$ (20)

 图 5 雅克比矩阵最大特征值与坡面角度的关系 Fig.5 Maximal absolute eigenvalue of the Jacobian as a function of the slope angle

 图 6 雅克比矩阵的特征值在复平面的空间分布图(坡面角度γ < 0.06) Fig.6 Eigenvalues of the Jacobian in the complex plane (slope angle γ < 0.06)

γ=0.009为例，长周期步态的周期解及雅克比矩阵的特征值分别为

 $\mathit{\boldsymbol{q}}{\rm{ = }}\left[ {\begin{array}{*{20}{c}} {{q_1}}\\ {{{\dot q}_1}}\\ {{q_2}}\\ {{{\dot q}_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {0.183\;3}\\ { - 0.210\;8}\\ { - 0.366\;5}\\ {0.043\;35} \end{array}} \right]$ (21)
 $\mathit{\boldsymbol{\lambda }} = \left[ {\begin{array}{*{20}{c}} {0.56 + 0.35{\rm{i}}}\\ {0.56 - 0.35{\rm{i}}}\\ 0\\ 0 \end{array}} \right]$ (22)

 $\mathit{q}{\rm{ = }}\left[ {\begin{array}{*{20}{c}} {{q_1}}\\ {{{\dot q}_1}}\\ {{q_2}}\\ {{{\dot q}_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {0.175\;4}\\ { - 0.215\;6}\\ { - 0.350\;9}\\ {0.040\;10} \end{array}} \right]$ (23)
 $\mathit{\lambda } = \left[ {\begin{array}{*{20}{c}} {2.15}\\ {0.69}\\ 0\\ 0 \end{array}} \right]$ (24)

 图 7 长周期步态和短周期步态的行走过程中模型的角度轨迹(γ=0.009) Fig.7 Motion trajectories of the long period solution and short period solution (γ=0.009)
3.2 全局稳定性

 图 8 不同初始状态的行走过程中模型的角度轨迹 Fig.8 Motion trajectories of the walking with different initial conditions
4 结论

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