Torque–stiffness-controlled dynamic walking with central pattern generators

Abstract

Walking behavior is modulated by controlling joint torques in most existing passivity-based bipeds. Controlled Passive Walking with adaptable stiffness exhibits controllable natural motions and energy efficient gaits. In this paper, we propose torque–stiffness-controlled dynamic bipedal walking, which extends the concept of Controlled Passive Walking by introducing structured control parameters and a bio-inspired control method with central pattern generators. The proposed walking paradigm is beneficial in clarifying the respective effects of the external actuation and the internal natural dynamics. We present a seven-link biped model to validate the presented walking. Effects of joint torque and joint stiffness on gait selection, walking performance and walking pattern transitions are studied in simulations. The work in this paper develops a new solution of motion control of bipedal robots with adaptable stiffness and provides insights of efficient and sophisticated walking gaits of humans.

Appendix

1.1 Appendix A: Lagrange’s equations for the dynamic walker

The model can be defined by the Euclidean coordinates $\mathbf {r}$, which can be described by the x-coordinate and y-coordinate of the center of mass of each stick and the corresponding directions.

The walker can also be described by the generalized coordinates $\mathbf {q}$:

$$\begin{aligned} \mathbf {q}=(x_h,y_h,\theta _1,\theta _2,\theta _b,\theta _{2s},\theta _{1f},\theta _{2f})^{'} \end{aligned}$$

(12)

We defined matrix $J$ as follows:

$$\begin{aligned} J = \mathrm{d}\mathbf {r}/\mathrm{d}\mathbf {q} \end{aligned}$$

(13)

The mass matrix in rectangular coordinate $\mathbf {r}$ is defined as:

$$\begin{aligned}&M = \mathrm{diag}(m_l, m_l, I_l, m_t, m_t, I_t, m_b, m_b, I_b, \nonumber \\&\qquad \qquad \qquad m_s, m_s, I_s, m_f, m_f, I_f, m_f, m_f, I_f) \end{aligned}$$

(14)

where m-components are the masses of each stick, while I-components are the moments of inertia, as shown in Fig. 1a.

The constraint function is marked as $\mathbf {\xi }(\mathbf {q})$, which is used to maintain foot contact with ground, the direction of the upper body and knee locking. Each component of $\mathbf {\xi }(\mathbf {q})$ should keep zero to satisfy the constraint conditions.

We can obtain the equations as following:

$$\begin{aligned} M_q\ddot{\mathbf {q}}&= \mathbf {F_q} + \varPhi ^{'}\mathbf {F_c}\end{aligned}$$

(15)

$$\begin{aligned} \mathbf {\xi }(\mathbf {q})&= \mathbf {0} \end{aligned}$$

(16)

where $\varPhi = \frac{\partial \mathbf {\xi }}{\partial \mathbf {q}}$. $\mathbf {F_c}$ is the constraint force vector. $M_q$ is the mass matrix in the generalized coordinates:

$$\begin{aligned} M_q = J^{'}MJ \end{aligned}$$

(17)

$\mathbf {F_q}$ is the active external force in the generalized coordinates:

$$\begin{aligned} \mathbf {F_q} = J^{'}\mathbf {F} - J^{'}M\frac{\partial J}{\partial \mathbf {q}}\dot{\mathbf {q}}\dot{\mathbf {q}} \end{aligned}$$

(18)

where $\mathbf {F}$ is the active external force vector in the Euclidean coordinates.

For the walking model in this paper, $\mathbf {F}$ includes gravitation, the damping torques, and the joint torques generated by the torsional springs. The sum of damping torques and compliance torques are calculated by Eq. (1). Thus, the natural dynamics of the model can be adjusted by controlling joint stiffness and equilibrium positions.

Equation (16) can be transformed to the followed equation:

$$\begin{aligned} \varPhi \ddot{\mathbf {q}} = -\frac{\partial (\varPhi \dot{\mathbf {q}})}{\partial \mathbf {q}}\dot{\mathbf {q}} \end{aligned}$$

(19)

Then the equations in matrix format can be obtained from Eqs. (15) and (19):

$$\begin{aligned} \left[ \begin{array}{cc} M_q &{} -\varPhi ^{'} \\ \varPhi &{} 0 \\ \end{array} \right] \left[ \begin{array}{c} \ddot{\mathbf {q}} \\ \mathbf {F_c}\\ \end{array} \right] = \left[ \begin{array}{c} \mathbf {F_q} \\ -\frac{\partial (\varPhi \dot{\mathbf {q}})}{\partial \mathbf {q}}\dot{\mathbf {q}} \\ \end{array} \right] \end{aligned}$$

(20)

The equation of the strike can be obtained by integration of Eq. (15):

$$\begin{aligned} M_q\dot{\mathbf {q}}^+ = M_q\dot{\mathbf {q}}^- + \varPhi ^{'}\varLambda _c \end{aligned}$$

(21)

where $\dot{\mathbf {q}}^+$ and $\dot{\mathbf {q}}^-$ are the generalized velocities just after and just before the strike, respectively. Here, $\varLambda _c$ is the impulse acted on the walker which is defined as follows:

$$\begin{aligned} \varLambda _c=\lim _{t^-\rightarrow t^+} \int _{t^-}^{t^+}\mathbf {F_c}\mathrm{d}t \end{aligned}$$

(22)

Since the strike is modeled as a fully inelastic impact, the walker satisfies the constraint function $\mathbf {\xi }(\mathbf {q})$. Thus, the motion is constrained by the followed equation after the strike:

$$\begin{aligned} \frac{\partial \mathbf {\xi }}{\partial \mathbf {q}} \dot{\mathbf {q}}^+ = \mathbf {0} \end{aligned}$$

(23)

Then the equation of strike in matrix format can be derived from Eqs. (21) and (23):

$$\begin{aligned} \left[ \begin{array}{cc} M_q &{} -\varPhi ^{'} \\ \varPhi &{} 0 \\ \end{array} \right] \left[ \begin{array}{c} \dot{\mathbf {q}}^+ \\ \varLambda _c\\ \end{array} \right] = \left[ \begin{array}{c} M_q\dot{\mathbf {q}}^- \\ \mathbf {0} \\ \end{array} \right] \end{aligned}$$

(24)

Parameters

$m_b = 12.0\,\mathrm{kg}$, upper body mass

$m_t = 2.5\,\mathrm{kg}$, thigh mass

$m_s = 2.5\,\mathrm{kg}$, shank mass

$m_f = 1.2\,\mathrm{kg}$, foot mass

$I_b = 0.36\,\mathrm{kg}\,\mathrm{m}^2$, moment of inertia of upper body

$I_t = 3.33\cdot 10^{-2}\,{\hbox {kg m}}^2$, moment of inertia of thigh

$I_s = 3.33\cdot 10^{-2}\,\hbox {kg m}^2$, moment of inertia of shank

$I_f = 4.0\cdot 10^{-3}\,\hbox {kg m}^2$, moment of inertia of foot

$l_b = 0.6\,\mathrm{m}$, upper body length

$l_t = 0.4\,\mathrm{m}$, thigh length

$l_s = 0.4\,\mathrm{m}$, shank length

$l_f = 0.2\,\mathrm{m}$, foot length

$r = 0.3$, foot ratio, which is defined as the ratio of the distance between the heel and the ankle to whole foot length

$g = 9.81\,\mathrm{ms}^{-2}$, gravitational acceleration

1.2 Appendix B: Parameters of central pattern generators

Parameters of oscillators for equilibrium position

The parameter values of an unit oscillator controlling the equilibrium position [as shown in Eq. (7)]:

$\tau _1 = \tau _2 = \tau _3 = \tau _4 = 0.02$,

$\tau _5 = \tau _6 = 0.05$.

$\tau ^{'}_1 = \tau ^{'}_2 = \tau ^{'}_3 = \tau ^{'}_4 = 0.01$,

$\tau ^{'}_5 = \tau ^{'}_6 = 0.02$.

$\beta = 0.005$.

The expressions of $\tilde{u}_i^e$ and $F_{\mathrm{eed},i}$ and the values of $w_{ij},\,c_i$ and $d_{ij}$ in each phase of different joints are listed in Table 2 (suppose leg $1$ is the stance leg).

Table 2 The parameters of the oscillators for equilibrium position control

It is worth mentioning that not all the terms of $w_{ij}$ and $d_{ij}$ are listed in the tables. The absent terms are taken to be zero. The column for feedback is not included in the table if there is no feedback at the corresponding joint. The knee joint of the stance leg in all the phases and the knee joint of the swing leg in phase $E,\,F,\, G$ and $H$ are locked, and the corresponding degrees of freedom are thus taken off, the parameters of joint $4$ in phase $E,\,F,\,G$ and $H$ and joint $3$ are not listed in the above tables.

Parameters of oscillators for stiffness

The parameter values of an unit oscillator controlling the joint stiffness [as shown in Eq. (8)]:

$\tau _i^s = 1,\,{\tau ^{'}}_i^s = 0.2,\quad i = 1, 2, \ldots , 6$

$\beta ^s = 0.02$.

The expressions of $\tilde{u}_i^s$ and $F_{\mathrm{eed},i}^s$ and the values of $w_{ij}^s,\,c_i^s$ and $d_{ij}^s$ in each phase of different joints are listed in Table 3 (suppose leg $1$ is the stance leg).

Table 3 The parameters of the oscillator for joint stiffness control

Similar to the case of equilibrium position control, not all the terms of $w_{ij}^s$ and $d_{ij}^s$ for stiffness control are listed in the tables. The absent terms are taken to be zero. The column for feedback is not included in the table if there is no feedback at the corresponding joint. Similarly, due to knee locking, the parameters of joint $4$ in phase $E,\, F,\, G$ and $H$ and joint $3$ are not listed in the above tables.