Inverse kinematics

Inverse kinematics#

Limitations of the conventional methods#

The main limitation of the conventional joint DOF angle calculation methods (e.g., dot product approach) is that the joint center of rotations is assumed to be the intersection of two vectors, and the derived joint angles might result in forward kinematics far from the original pose. On the other hand, the optimization-based inverse kinematics approaches, given the kinematic chain of the links (i.e., rotational axes and pivot points of joints), try to match the end effector of the kinematic chain to the real end effector as closely as possible. However, this method disregards the position of the previous joints, leading to unrealistic forward kinematics.

Sequential inverse kinematics#

To reconcile the shortcomings of the two methods above, we developed a sequential inverse kinematics calculation algorithm, also known as body movement optimization. This approach is inspired by the control of humanoid robots based on human motion capture data [BAD18].

Specifically, our method begins with the proximal-most leg segment to calculate certain DOF angles of the subsequent joint. It then uses the angles calculated in the previous step and extends the kinematic chain by adding another segment, repeating this process until the tip of the chain is reached. This approach comprises four steps as follows:

  • Stage 1: kinematic chain consists of the coxa leg segment and is used to calculate Thorax-Coxa yaw (i.e., rotation around the anteroposterior axis) and pitch (i.e., rotation around the mediolateral axis) by following the tip of coxa as an end-effector

  • Stage 2: kinematic chain consists of the coxa and the trochanter + femur (fused) leg segments and is used to calculate the Thorax-Coxa roll (i.e., rotation around the dorsoventral axis), Coxa-Trochanter pitch (i.e., rotation around the mediolateral axis) by following the tip of the femur as an end-effector

  • Stage 3: kinematic chain consists of the coxa, the trochanter & femur (fused), and tibia leg segments and is used to calculate the Trochanter-Femur roll (i.e., rotation around the dorsoventral axis), Femur-Tibia pitch (i.e., rotation around the mediolateral axis) by following the tip of the tibia as an end-effector

  • Stage 4: The kinematic chain consists of the entire leg segments and is used to calculate the Tibia-Tarsus pitch (i.e., rotation around the mediolateral axis) by following the tip of the leg (i.e., Claw) as an end-effector

Our pipeline builds upon the open-source inverse kinematics library IKPy, which uses an optimizer from SciPy to minimize a scalar function, that is, the Euler distance between the original pose and the forward kinematics from the calculated joint angles.

Note

When replaying the joint angles in simulation, it is essential to keep the kinematic chain of the biomechanical model consistent with the kinematic chain used to calculate the joint angles.

For instance, if the thorax-coxa joint angles are calculated in the order of yaw-pitch-roll, then the biomechanical model should also have the thorax-coxa joint DOFs in the same order since we use the intrinsic Euler convention to calculate the joint angles.

Head and antennal joint angle calculation#

Since the head consists of two coupled kinematic chains (left and right antennae) attached to the same neck joint, a single kinematic chain formulation is not suitable, as it tends to bias the solution toward one antenna and introduces angle estimation errors. We therefore compute head and antennal joint angles using vector-based geometric definitions instead of a unified inverse kinematics model.

The vectors constructed for the head joint angles are as follows:

  • Head roll: The angle between the vector from the right antenna base to the left antenna base, and the global mediolateral axis in the transverse plane.

  • Head pitch: The angle between the vector from the neck to the mid-antennae base and the global anteroposterior axis in the sagittal plane. We subtracted the resting head pitch angle from the calculated joint angles to obtain joint angles relative to the resting position.

  • Head yaw: The angle between the vector from the right antenna base to the left antenna base, and the global anteroposterior axis in the dorsal plane.

  • Antennal pitch: The angle between the vector from the neck to the antenna base and the vector from the antenna base to the antenna tip in the sagittal plane.

  • Antennal yaw: The angle between the vector from the right antenna base to the left antenna base and the vector from the antenna base to the antenna tip in the transverse plane.

Note that, when the head rotation reaches 90 degrees, the antennal pitch and yaw calculations will swap their roles, causing inaccurate computations. To circumvent this issue, we first calculate the head joint angles and then de-rotate the head key points by the amount of head rotation to calculate the antennal joint angles.

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