The multilegged walking robots benefit from their complex morphology when traversing rough terrains. However, reliable foot contact detection is required to exploit their locomotion capabilities fully. Based on a dynamic model of the leg movements, foot-contact detection is possible using position feedback only. Since dynamic model determination can be demanding and laborious, we propose employing spare identification of nonlinear dynamics to construct the closed-form contact-free leg dynamics model without explicit manual identification. Model predictions and current measurements are then used to detect deviations from contact-free leg dynamics and thus determine the leg contact with an obstacle or terrain. The feasibility of the proposed approach is validated and compared with a precise high-fidelity analytical model.
In rough terrain locomotion with multi-legged robots,
the crucial part of locomotion control is a timely and
reliable sense of the leg contact with the terrain or
obstacles. It is specifically essential for position-based leg
control, where the internal model of leg dynamics can be
utilized to estimate the foot-contact [
Therefore, the dynamics model needs to be adjusted to such changes to support reliable contact sensing.
The model-based contact detection method can be
based on an inverse dynamics model to estimate contact
force [
Hence, it might be cumbersome [
On the other hand, machine learning-based approaches estimate input-output relation directly from the training data, including phenomena omitted by the analytical models. However, a black-box-based machine learning approach might result in a physically infeasible model.
Therefore, using physics-informed machine learning can be advantageous to overcome the laborious parameter identification yet have a “gray-box” model that can fit the properties of the handcrafted models. Furthermore, we can generalize the approach toward an online learnable system that adapts to non-stationarities and changes in the system caused by external factors.
We propose to develop a lightweight learning-based
physics-informed contact detection method using only
position feedback from the servomotors. In [
The remainder of the paper is organized as follows.
The addressed problem of foot contact detection is stated in Section 2. The proposed detection method is described in Section 3. Experimental evaluation results are reported in Section 4. Concluding remarks are summarized in Section 5.
In multi-legged locomotion, the robot legs are moved in
repetitive patterns defined by the utilized gait.
Therefore, the particular leg trajectory can be divided into gait
segments based on the trajectory shape. For foot contact
detection, the most relevant segments are those in which
we expect the contact of the leg with the terrain. Such
segments are referred to as contact (gait) segments. In the
presented work, we consider a 3-DoF leg consisting of
three servomotors references as coxa, femur, and tibia
and the whole class of trajectories when the foot-tip
descends from the upper sub-space of the workspace to the
lower one.
2. Problem We aim at learning-based physics-informed contact
detection; hence, we propose to utilize the Sparse
IdentiThe multi-legged robot locomotion can be based on ifcation of Nonlinear Dynamical systems (SINDy) [
For the used robotic platform depicted in Fig. 1, two needed to determine the function (()). These time data types are available to model the robot dynamics at histories are organized into respective rows of the and any given discrete time-step and each -th joint. The ˙ matrices. data are the measured joint positions real[] and desired Furthermore, a candidate library Θ() is an esjoint positions des[] set by the locomotion controller. sential part of the concept as it consists of candiThe measured and desired joint positions can be recorded date non-linearities applied to the system stated. For for and ′ past time steps, respectively. Furthermore, example, the relevant functions for our setup are each time step, future time steps of the desired joint the identity and goniometric functions Θ() = positions are available. [ sin() cos()]. Then, the sparse vector of co
Hence, we formulate the foot-contact detection as a efiecients Ξ = [ 1 2 · · · ] is to be determined using time-series prediction of the -th joint position pred[+1] sparse-promoting algorithm from the current and historical data pred[ + 1] = ˜( real[ − ], . . . , real[], des[ − ′], . . . , des[], . . . , des[ + ]), (1)
The control input () to the system dynamics
˙ = Θ()Ξ.
˙() = ((), ()),
(3)
(4)
where real[] denotes the measured joint positions vector
of all joints coupled with the -th joint. Similarly, we
can be incorporated into the model using the approach
by E. Kaiser et al. [
(5) By defining the system state vector [] and the input vector [] as [] = [ real[ − ] · · · real[]] the function ˜ from (1) can be found using the SINDy.
For the considered robotic platform, the coupling efect
between each pair of legs is below the resolution of the
utilized servomotors [
ifdelity model of a real hexapod robot [
The model has been learned using datasets collected within the simulation environment; see Section 4.1. Then, we study the influence of the model parameters on the learned model, and the results are reported in Section 4.2. Finally, the contact detection results are presented in Section 4.3.
Having the learned model (function ˜), the foot con- In the collected dataset, the selected hexapod leg follows
tact is based on the adaptive thresholding as in [
50 20 30
Time step [-]
An example of the collision-free trajectory for the second joint (femur servomotor) is depicted in Fig. 2. It can be observed that the prediction accuracy might vary within the selected operational space. Hence, using a ifxed threshold value might be challenging. Therefore, we opt for using the absolute prediction error diference between two consecutive predictions of selected joint subsets and experimentally found threshold values to detect a contact of the particular leg with the terrain.
The Python programming language module PySindy [
(8) where 2 represents the polynomial features up to the second-degree polynomial, including the constant term.
We have selected a universal Sequentially Thresholded
Least Squares (STLSQ) algorithm optimizer [
The search is stopped when the threshold causes the leg dynamics in foot contact with the terrain. The proelimination of all coeficients for any state variable. Note posed physics-informed lightweight learning approach that the achieved scores are comparable regardless of the has been evaluated to detect the leg contact with the particular threshold. Therefore, the inner model struc- terrain in locomotion control of a small hexapod walkture has been examined to consider the sensitivity to ing robot with only position feedback. Based on the the control input des[] since the accuracy of the naive high-fidelity simulation, the proposed approach is viapproach where the prediction is the current position tal, and for a given workspace subspace, the proposed pred[+1] = real[] is significantly close to the accuracy method can detect contact. However, we plan to employ of the models regardless of the threshold. Based on the model ensembling to extend the method to the whole presented results, we select = 5 parametrization for leg’s workspace. the future evaluation of the model in contact detection. training data to evaluate foot contact detection of the proposed method. During the trajectory execution, real joint positions real and desired joint positions des have been collected at = 50 Hz. Besides, the robot body orientation and selected leg servomotor torques are collected at the highest possible sampling rate of = 1000 Hz to provide ground truth measurements for the foot contact.
The plots of the robot orientation, torques of the moving leg’s joints, and real and predicted joints’ position errors are depicted in Fig. 5. The results support the hypothesis that the leg contact with the terrain can be detected using absolute prediction error diference between two consecutive predictions.
The work was supported by the Czech Science Foundation (GAČR) under research project No. 21-33041J. r u :05 ;efm ]
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