Hexapod Leg Inverse Kinematics: 3-DOF Cycloidal Step and Workspace

The geometry follows Meisner's geometric hexapod IK derivation and pinned worked notebook (attached sources s1, s2 and s5, page 1). The implementation uses quadrant-aware atan2 and signed joint bends. With $r=\sqrt{x^2+y^2}$, $d^2=(r-L_c)^2+z^2$ and $D=(d^2-L_f^2-L_t^2)/(2L_fL_t)$:

The knee branch has $s=-1$ or $+1$. Reachability is checked before accepting results; the acos argument is clamped only to keep rejected configurations and roundoff from causing domain errors. Forward kinematics independently checks the complete sampled path.

The conventional composite cycloid is the baseline in Spider-Leg-Inspired Structural Design and Bézier Foot Trajectory Planning for Stable Walking of a Hexapod Robot (2026), DOI 10.3390/biomimetics11050352, section on composite cycloid trajectories. For $u=t/T$, the distance along the step is $S[u-\sin(2\pi u)/(2\pi)-1/2]$ and height is $z_0+H[1-\cos(2\pi u)]/2$. Analytic derivatives supply Cartesian velocity.

The translational Jacobian is differentiated from $r=L_c+L_f\cos q_2+L_t\cos(q_2+q_3)$ and $z=L_f\sin q_2+L_t\sin(q_2+q_3)$. Joint rates solve $J\dot q=\dot p$ exactly, without damping. Yoshikawa (1985), DOI 10.1177/027836498500400201, defines $w=\sqrt{\det(JJ^T)}=|\det J|$. Here $w=rL_fL_t|\sin q_3|$. The color maps divide this by the geometric upper bound $(L_c+L_f+L_t)L_fL_t$, so color lies between zero and one. This normalization is separate from Yoshikawa's dimensional measure.

The eigenvectors of $JJ^T$ give the ellipsoid directions, and square roots of its eigenvalues give semi-axes (Lynch and Park, Modern Robotics, 2017, §5.4). The yaw direction decouples from the radial–vertical block, whose two eigenvalues are solved analytically. Isotropy is $\sigma_{min}/\sigma_{max}$. The three displayed semi-axes are sagittal major, sagittal minor and tangential; their numerical ordering can change.

Rigid links and ideal revolute joints; the coxa offset is horizontal, the femur and tibia pitch axes are parallel, and the fixed hip frame does not translate during swing. Joint limits are geometric angles, with no servo offsets. IK selects the yaw pointing toward the foot and one continuous knee branch. Other configurations with the coxa pointing away from the foot are not included.

The velocity ellipsoid uses equal joint-rate weighting: $\|\dot q\|_2=1$ rad/s. Displayed lengths in mm/rad become foot velocities in mm/s under that normalization. This sphere is not the box formed by three independent actuator speed limits. Defaults are an illustrative small robot geometry, not a particular commercial leg.

The tool evaluates 601 evenly spaced swing poses. Each workspace color mask is evaluated directly on a 256 × 256 grid, with excluded configurations kept empty. The outlines follow 721-point joint-limit curves; horizontal radial boundaries are located by bisection after a 1025-point scan, and yaw-sector edges are straight. Isocontours use a 101 × 81 radial–height grid and a 121 × 121 horizontal grid. Boundary detail below the display grid spacing and very narrow permitted or excluded radial intervals below the scan spacing can be missed. Swing extrema and limit margins are sampled estimates: very narrow limit violations or speed peaks between samples can be missed. Add margin when selecting actuators or paths near a boundary. Residual summaries round below 10⁻⁹ mm or mm/s to remove floating-point noise.

The radial map is independent of yaw and applies only within the entered yaw sector; the horizontal map includes all three joint limits. Its path overlay is the horizontal projection of the lifted path. The workspace maps show the full path and remain static during playback; the leg and velocity ellipsoid animate.

The conventional cycloid has zero endpoint velocity but nonzero vertical acceleration at lift-off and touchdown. Joining it directly to a stationary stance gives a vertical acceleration discontinuity; it is not a jerk-limited or impact-free contact model. Animation shows one swing and stops at touchdown. Playback uses a fixed display rate; reported velocities use the entered swing duration.

At a singularity, arbitrary Cartesian velocity has no unique finite joint-rate solution; singular pitch-rate samples on a moving trajectory are left undefined, including lift-off and touchdown where instantaneous foot velocity is zero. Finite one-sided limits can exist, but this Jacobian-inverse calculation does not substitute a trajectory limit for an undefined sample. Peak pitch speeds and the speed-only minimum duration are also undefined when any sampled pitch rate is undefined. Zero joint rates at a singular pose are assigned only when both step length and clearance are zero, making the entire trajectory stationary. Near singularities, a small foot perturbation may demand a large joint-speed change. Speed-only minimum duration is meaningful only for a reachable, continuous path satisfying joint-angle limits; it does not check torque or acceleration. The common speed limit is applied to each joint separately.