Toward an Expressive Bipedal Robot: Variable Gait Synthesis and Validation in a Planar Model


Humans are efficient, yet expressive in their motion. Human walking behaviors can be used to walk across a great variety of surfaces without falling and to communicate internal state to other humans through variable gait styles. This provides inspiration for creating similarly expressive bipedal robots. To this end, a framework is presented for stylistic gait generation in a compass-like under-actuated planar biped model. The gait design is done using model-based trajectory optimization with variable constraints. For a finite range of optimization parameters, a large set of 360 gaits can be generated for this model. In particular, step length and cost function are varied to produce distinct cyclic walking gaits. From these resulting gaits, 6 gaits are identified and labeled, using embodied movement analysis, with stylistic verbs that correlate with human activity, e.g., “lope” and “saunter”. These labels have been validated by conducting user studies in Amazon Mechanical Turk and thus demonstrate that visually distinguishable, meaningful gaits are generated using this framework. This lays groundwork for creating a bipedal humanoid with variable socially competent movement profiles.

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This work was conducted under IRB #17697 and funded by National Science Foundation (NSF) Grant #1701295. The authors would like to thank Prof. Hae Won Park for useful discussions about the controller design and trajectory optimization and Prof. Joshua Schultz for useful discussions about how this control scheme might be implemented through a physical mechanism.

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Correspondence to Umer Huzaifa.

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A LaViers owns stock in AE Machines, an automation software company.

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The modeling matrices of Eq. 2 are as follows:

$$\begin{aligned} D_s = \begin{bmatrix} \left( \frac{5}{4}m\right) r^2&\quad - \frac{m}{2}r^2c_{12} \\ - \frac{m}{2}r^2c_{12}&\quad \frac{m}{4}r^2 \\ \end{bmatrix}, \end{aligned}$$

where \(c_{12}=\cos (q_{st}-q_{sw})\) and

$$\begin{aligned} C_s= \begin{bmatrix} 0&\quad -\frac{m}{2}r^2\dot{q}_{sw}s_{12} \\ \frac{m}{2}r^2\dot{q}_{st} s_{12}&\quad 0 \\ \end{bmatrix}, \end{aligned}$$

where \(s_{12} = \sin (q_{st}-q_{sw}))\) and

$$\begin{aligned} G_s= \begin{bmatrix} -(\frac{3m}{2})gr\sin (q_{st}) \\ \frac{m}{2}gr \sin (q_{sw}))\\ \end{bmatrix}, \end{aligned}$$

The matrices from Eq. 5 are as follows:

$$\begin{aligned} D_e= \begin{bmatrix} D_s&\quad D_{12} \\ D^T_{12}&\quad D_{22} \\ \end{bmatrix} \end{aligned}$$
$$\begin{aligned} D_{12}= \begin{bmatrix} \frac{3m}{2}r\cos q_{st}&\quad -\frac{3m}{2}r\sin q_{st}\\ -\frac{m}{2}r\cos q_{sw}&\quad \frac{mr}{2}\sin q_{sw} \\ M_t&0 \end{bmatrix} \end{aligned}$$


$$\begin{aligned} D_{22}= \begin{bmatrix} 2m&\quad 0\\ 0&\quad 2m \end{bmatrix}. \end{aligned}$$


$$\begin{aligned} C_e= \begin{bmatrix} C_s&\quad 0_3\\ C_1&\quad 0_2 \end{bmatrix}, \end{aligned}$$


$$\begin{aligned} C_1= \begin{bmatrix} -\frac{3m}{2} r \dot{q}_{st} \sin q_{st}&\quad \frac{mr}{2} \dot{q}_{sw} \sin q_{sw} \\ -\frac{3m}{2} r \dot{q}_{st} \cos q_{st}&\quad \frac{mr}{2} \dot{q}_{sw} \cos q_{sw} \\ \end{bmatrix}, \end{aligned}$$


$$\begin{aligned} G_e= \begin{bmatrix} G_s\\ G_1\\ \end{bmatrix}, \end{aligned}$$


$$\begin{aligned} G_1= \begin{bmatrix} 0\\ 2mg\\ \end{bmatrix}. \end{aligned}$$

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Huzaifa, U., Maguire, C. & LaViers, A. Toward an Expressive Bipedal Robot: Variable Gait Synthesis and Validation in a Planar Model. Int J of Soc Robotics 12, 129–141 (2020).

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  • Biped locomotion
  • Human-like natural motions
  • Stylistic motion variation synthesis
  • Expressivity
  • Optimization
  • Embodied movement analysis