Spinal joint compliance and actuation in a simulated bounding quadruped robot

Abstract

Spine movements play an important role in quadrupedal locomotion, yet their potential benefits in locomotion of quadruped robots have not been systematically explored. In this work, we investigate the role of spinal joint actuation and compliance on the bounding performance of a simulated compliant quadruped robot. We designed and conducted extensive simulation experiments, to compare the benefits of different spine designs, and in particular, we compared the bounding performance when (i) using actuated versus passive spinal joint, (ii) changing the stiffness of the spinal joint and (iii) altering joint actuation profiles. We used a detailed rigid body dynamics modeling to capture the main dynamical features of the robot. We applied a set of analytic tools to evaluate the bounding gait characteristics including periodicity, stability, and cost of transport. A stochastic optimization method called particle swarm optimization was implemented to perform a global search over the parameter space, and extract a pool of diverse gait solutions. Our results show improvements in bounding speed for decreasing spine stiffness, both in the passive and the actuated case. The results also suggests that for the passive spine configuration at low stiffness values, periodic solutions are hard to realize. Overall, passive spine solutions were more energy efficient and self-stable than actuated ones, but they basically exist in limited regions of parameter space. Applying more complex joint control profiles reduced the dependency of the robot’s speed to its chosen spine stiffness. In average, active spine control decreased energy efficiency and self-stability behavior, in comparison to a passive compliant spine setup.

Appendix: Extracting equations of motion

The equations of motion can be derived based on Lagrange method, through the following steps. First, the position vector of the center of mass of the front/hind trunk segments can be obtained as:

$$\begin{aligned} r_{B_f}= & {} [x+x_{O_f}, y+y_{O_f}, 0]^T \end{aligned}$$

(16)

$$\begin{aligned} r_{B_h}= & {} [x+x_{O_h}, y+y_{O_h}, 0]^T \end{aligned}$$

(17)

where subscripts f and h present front and hind segments. Similarly, we require the COM position vector for the leg’s upper segments (shown with index U):

$$\begin{aligned} r_{U_F}= & {} \left[ \begin{array}{@{}r@{\quad }cclr@{}} &{} x + l_{U_f} \sin (\theta _f) + x_{O_f} &{}\\ &{} y - l_{U_f} \cos (\theta _f) + y_{O_f} &{}\\ &{} 0 &{} \end{array} \right] \end{aligned}$$

(18)

$$\begin{aligned} r_{U_H}= & {} \left[ \begin{array}{@{}r@{\quad }cclr@{}} &{} x + l_{U_h} \sin \theta _h + x_{O_h} &{}\\ &{} y - l_{U_h} \cos \theta _h + y_{O_h} &{}\\ &{} 0 &{} \end{array} \right] \end{aligned}$$

(19)

and leg’s lower segments (shown with index L):

$$\begin{aligned} r_{L_f}= & {} \left[ \begin{array}{@{}r@{\quad }cclr@{}} &{} x + (l_f - l_{L_f}) \sin \theta _f + x_{O_f} &{}\\ &{} y - (l_f - l_{L_f}) \cos \theta _f + y_{O_f} &{}\\ &{} 0 &{} \end{array} \right] \end{aligned}$$

(20)

$$\begin{aligned} r_{L_H}= & {} \left[ \begin{array}{@{}r@{\quad }cclr@{}} &{} x + (l_h - l_{L_h}) \sin \theta _h + x_{O_h} &{}\\ &{} y - (l_h - l_{L_h}) \cos \theta _h + y_{O_h} &{}\\ &{} 0 &{} \end{array} \right] \end{aligned}$$

(21)

where $\theta _i = \alpha _i + \psi _i$ for $i \in \left\{ f,h \right\}$, $\psi _i$ and $\alpha _i$ respectively represent trunk pitch angle and the leg’s hip angle. The vector $[x_{O_f}, y_{O_f}]$ shows the position of the connection point of the front leg to the front trunk segment and can be written as:

$$\begin{aligned} \left\{ \begin{array}{@{}r@{\quad }cclr@{}} &{} x_{O_f} = l_{1_f} \cos \psi _f \\ &{} y_{O_f} = l_{1_f} \sin \psi _f \end{array} \right. \end{aligned}$$

(22)

Similarly, for the hind leg we have:

$$\begin{aligned} \left\{ \begin{array}{@{}r@{\quad }cclr@{}} &{} x_{O_h} = - l_{1_h} \cos \psi _h \\ &{} y_{O_h} = - l_{1_h} \sin \psi _h \end{array} \right. \end{aligned}$$

(23)

Now, the COM velocities for the trunk segment can be obtained as the derivative of Eqs. 16 and 17:

$$\begin{aligned} \mathbf v _{B_i} = \left[ \begin{array}{@{}r@{\quad }cclr@{}} &{} \dot{x} - l_{1_i} \dot{\psi _i} \cos \psi _i &{}\\ &{} \dot{y} + l_{1_i} \dot{\psi _i} \sin \psi _i &{}\\ &{} 0 &{} \end{array} \right] \end{aligned}$$

(24)

with i being F for the front and H for the hind trunk segments. Similarly, for the leg upper segment COM velocities, we obtain:

$$\begin{aligned} \mathbf v _{U_i} = \left[ \begin{array}{@{}r@{\quad }cclr@{}} &{} \dot{x} + l_{2_i} \dot{\theta _i} \cos \theta _i - l_{1_i} \dot{\psi _i} sin(\psi _i) &{}\\ &{} \dot{y} + l_{2_i} \dot{\theta _i} \sin \theta _i + l_{1_i} \dot{\psi _i} sin(\psi _i)&{}\\ &{} 0 &{} \end{array} \right] \end{aligned}$$

(25)

and for the leg lower segment COM velocities:

$$\begin{aligned} \mathbf v _{L_i} = \left[ \begin{array}{@{}r@{\quad }cclr@{}} &{} \dot{x} - (l_i - l_{3_i} \dot{\theta _i} \cos \theta _i + \dot{l}_i \sin \theta _i - l_{1_i} \dot{\psi _i} \sin \psi _i) &{}\\ &{} \dot{y} - (l_i - l_{3_i} \dot{\theta _i} \sin \theta _i - \dot{l}_i \sin \theta _i + l_{1_i} \dot{\psi _i} \cos \psi _i) &{}\\ &{} 0 &{} \end{array} \right] \end{aligned}$$

(26)

Furthermore, we require rotational velocities for different segments. It can be written for the trunk segment as follows:

$$\begin{aligned} \varvec{\varOmega }_{m_i} = [0, ~0, ~\dot{\psi _i}]^T \end{aligned}$$

(27)

and for the leg upper/lower segments:

$$\begin{aligned} \varvec{\varOmega }_{U_i} = \varvec{\varOmega }_{L_i} = [0, ~0, ~\dot{\theta _i}]^T \end{aligned}$$

(28)

Now, by using the COM position and velocity vectors, we can provide the kinetic energy of the system:

$$\begin{aligned} T = \sum \limits _{i=1}^n \left( \dfrac{1}{2} m_i \dot{r}_i^T \dot{r}_i + \dfrac{1}{2} I_i \varOmega _i^T \varOmega _i\right) \end{aligned}$$

(29)

Based on the Lagrange formulation, the mass matrix can be derived via the following partial derivation of the kinetic energy:

$$\begin{aligned} M = \dfrac{\delta \left( \frac{\delta T}{\delta \dot{q}}\right) }{\delta \dot{q}} \end{aligned}$$

(30)

The centrifugal and Coriolis terms can be derived as:

$$\begin{aligned} b = \dfrac{\delta \left( \frac{\delta T}{\delta \dot{q}}\right) }{\delta \dot{q}} \dfrac{dq}{dt} - \dfrac{dT}{dq} \end{aligned}$$

(31)

The gravity term can be derived as the partial derivative of the potential energy to the general coordinates:

$$\begin{aligned} g = \dfrac{\delta V}{\delta q}^T \end{aligned}$$

(32)

and the potential energy can be obtained as:

$$\begin{aligned} V = \sum \limits _{i=1}^n \left( r_i^T F_i^g\right) \end{aligned}$$

(33)

where $F_i^g$ is the gravitational force exerted on segment i:

$$\begin{aligned} F_i^g = [0,~ -m_ig,~0]^T \end{aligned}$$

(34)

To simulate the system dynamics, one requires matrices M, b, and g from Eqs. 30, 31 and 32. However, this system is a hybrid dynamical system, where the external forces vary during flight and stance phases. To account for the discontinuous dynamics, the contact forces on the right hand side of the Eq. 2 should be updated per phase. To do so, we first extract the contact point position vector as:

$$\begin{aligned} r_{c_i} = \left[ \begin{array}{@{}r@{\quad }cclr@{}} &{} x + x_{O_i} + l_i \sin \theta _i &{}\\ &{} y + y_{O_i} - l_{i} \cos \theta _H - r_{foot} &{}\\ &{} 0 &{} \end{array} \right] \end{aligned}$$

(35)

where $i \in \left\{ f,h \right\}$, and $r_{foot}$ is the foot radius. The contact Jacobian can then be derived as:

$$\begin{aligned} J_c = \left[ \dfrac{d r_{c_F}}{d q}, \dfrac{d r_{c_H}}{d q}\right] \end{aligned}$$

(36)

and the contact force can be computed as follows:

$$\begin{aligned} F_c = J_c M^{-1} J_c^T \left\{ J_c M^{-1}(b+g-\tau _{act}) \right\} \end{aligned}$$

(37)

For more details please see Remy (2011).