Minxing Sun

Estimator.git, W+rw​Δ​ψi,keff​𝐭^k,\mathbf{c}^{W}_{i,k}\leftarrow\mathbf{c}^{W}_{i,k-1}+r_{w}\,\Delta\psi^{\mathrm{eff}}_{i,k}\,\hat{\mathbf{t}}_{k}, (23) where rwr_{w} is the wheel radius (parameter Par_WheelRadius). This step is applied only when the leg is deemed in contact, and it reduces systematic bias that would arise if a rolling wheel were treated as a fixed stance anchor. Velocity augmentation. The same idea is used to augment the stance-based velocity observation. In implementation we subtract the joint-induced shank pitch rate (q˙2+q˙3)(\dot{q}_{2}+\dot{q}_{3}); including θ˙pitch\dot{\theta}_{\mathrm{pitch}} is straightforward but was found unnecessary in our runs. Using the encoder angular velocity and joint angular velocities, we form an effective wheel rolling rate ψ˙i,keff=ψ˙i,k−(q˙i,2,k+q˙i,3,k),\dot{\psi}^{\mathrm{eff}}_{i,k}=\dot{\psi}_{i,k}-\left(\dot{q}_{i,2,k}+\dot{q}_{i,3,k}\right), (24) consistent with the implementation (wheel rate minus shank pitch rate induced by the leg joints). The induced planar rolling velocity is 𝐯roll,i,kW=rw​ψ˙i,keff​𝐭^k,\mathbf{v}^{W}_{\mathrm{roll},i,k}=r_{w}\,\dot{\psi}^{\mathrm{eff}}_{i,k}\,\hat{\mathbf{t}}_{k}, (25) which is added to the kinematics-derived stance velocity term when forming the fused body velocity observation (cf. (8)). Degenerate case and point-foot platforms. If rw=0r_{w}=0, the propagation (23)–(25) vanishes and the formulation reduces to the point-foot case where the stance anchor is treated as stationary. In our parameterization, point-foot robots absorb the foot radius into the effective shank length (Section III-A), whereas wheel-legged robots keep rw>0r_{w}>0 and apply the explicit rolling propagation above. IV Inverse-Kinematics Cubature Kalman Filtering for Foot-End Velocity Contact-anchored proprioceptive odometry (Section II) relies on stance-phase kinematics to provide body-velocity observations. A practical challenge is that joint angular velocities obtained from motor encoders are affected by quantization and discrete-time differentiation, and their noise is amplified by the nonlinear leg kinematics. As a result, the derived hip-to-foot velocity can exhibit impulsive spikes, which then propagate into the fused body velocity and degrade the stability of contact-anchored updates. To suppress these artifacts while keeping the pipeline purely proprioceptive, we introduce an IKVel-CKF that estimates the hip-to-end-effector Cartesian velocity directly from joint angles and joint velocities. The analytic inverse-kinematics mapping and the CKF recursion used in our implementation are provided in Appendix B. IV-A Motivation: Spurious Velocity Spikes from Encoder-Derived Joint Rates Figure 7 (left) shows the estimated body linear velocity when the system uses raw encoder-derived joint rates without IKVel-CKF. The horizontal channels (vxv_{x} and vyv_{y}) contain pronounced spikes, reaching up to roughly 200%200\% of the nominal signal magnitude. These spikes are consistent with encoder quantization, discrete differentiation, and slight timing mismatch, which are then magnified by nonlinear kinematic projection. The issue is further illustrated at the leg level in Fig. 8, where the raw hip–foot velocity of a representative leg (blue) contains impulsive excursions. IV-B IKVel-CKF: Nonlinear Filtering under an Inverse-Kinematics Measurement Model For each leg ii, IKVel-CKF maintains a, D loop) Figure 18 reports representative real-time estimated (x,y,z)(x,y,z) traces for Robot A and Robot B. Each robot executes one closed-loop trajectory that combines approximately ∼\sim200 m of horizontal travel with an accumulated ∼\sim15 m vertical excursion, returning to the start. Robot A achieves 0.16380.1638 m horizontal-loop error and 0.2190.219 m vertical-loop error, while Robot B yields 0.22640.2264 m and 0.1990.199 m, respectively (Table II). (a) Astrall A (point-foot): ∼\sim200 m horizontal loop and ∼\sim15 m vertical loop. (b) Astrall B (wheel-legged): ∼\sim200 m horizontal loop and ∼\sim15 m vertical loop. Figure 18: Astrall closed-loop trials: real-time estimated (x,y,z)(x,y,z) traces for Robot A (MP) and Robot B (MW). Robot C (two separate trials) Robot C is evaluated in two separate runs: (i) a substantially longer ∼\sim700 m horizontal closed loop, and (ii) a ∼\sim20 m vertical loop trajectory. Figure 19 reports the corresponding real-time estimated (x,y,z)(x,y,z) traces. The horizontal-loop error increases to 7.687.68 m, while the vertical-loop error remains 0.5400.540 m (Table II). We attribute the larger horizontal drift primarily to slip events and contact-geometry violations that are more pronounced for long-range wheel-legged operation, especially during step-down segments where the motion may involve brief loss of sustained rolling contact. (a) Astrall C: ∼\sim700 m horizontal closed loop. (b) Astrall C: ∼\sim20 m vertical closed loop. Figure 19: Astrall Robot C closed-loop trials: real-time estimated (x,y,z)(x,y,z) traces. Public artifacts. To support reproducible evaluation, we release (i) the complete implementation and (ii) the ROS bags of the Unitree Go2 EDU trials, including synchronized video.111https://github.com/ShineMinxing/Ros