Multi-contact vertical ladder climbing with an HRP-2 humanoid

Abstract

We describe the research and the integration methods we developed to make the HRP-2 humanoid robot climb vertical industrial-norm ladders. We use our multi-contact planner and multi-objective closed-loop control formulated as a QP (quadratic program). First, a set of contacts to climb the ladder is planned off-line (automatically or by the user). These contacts are provided as an input for a finite state machine. The latter builds supplementary tasks that account for geometric uncertainties and specific grasps procedures to be added to the QP controller. The latter provides instant desired states in terms of joint accelerations and contact forces to be tracked by the embedded low-level motor controllers. Our trials revealed that hardware changes are necessary, and parts of software must be made more robust. Yet, we confirmed that HRP-2 has the kinematic and power capabilities to climb real industrial ladders, such as those found in nuclear power plants and large scale manufacturing factories (e.g. aircraft, shipyard) and construction sites.

Appendix: Computing orientation task \(\mathrm {Err}\)

To compute the task orientation error $\text{ Err }$ between two rotation matrices $E_1$ and $E_2$, we use the logarithmic matrix formula described in Murray et al. (1994).

Let $E = E_1^T E_2$, ${\hat{\omega }} \in {\mathbb {R}}^{3\times 3}$ the skew matrix representing angular velocities, and the rotational speed matrix ${\dot{E}} = {\hat{\omega }} E$,

$$\begin{aligned} E = \begin{bmatrix} E_{11}&E_{12}&E_{13} \\ E_{21}&E_{22}&E_{23} \\ E_{31}&E_{32}&E_{33} \end{bmatrix} \end{aligned}$$

We have $\log (\text{ E }) = \log (\exp ({\hat{\omega }})) \equiv {\hat{\omega }}$. To compute the rotational speed vector $\omega \in {\mathbb {R}}^{3}$ we use the following formula:

$$\begin{aligned} \omega = \frac{1}{2\mathrm {sinc}(\theta )} \begin{bmatrix} E_{32} - E_{23}\\ E_{13} - E_{31}\\ E_{21} - E_{12} \end{bmatrix} \end{aligned}$$

where $\theta = \cos ^{-1}\left( \frac{E_{11} + E_{22} + E_{33} - 1}{2} \right)$. Then we simply set:

$$\begin{aligned} \text{ Err }(E_1, E_2) = \omega \end{aligned}$$