Redundancy parameterization and inverse kinematics of 7-DOF revolute manipulators

Abstract

Seven degree-of-freedom (DOF) robot arms have one redundant DOF which does not change the motion of the end effector. The redundant DOF offers greater manipulability of the arm configuration to avoid obstacles and singularities, but it must be parameterized to fully specify the joint angles for a given end effector pose. For 7-DOF revolute (7R) manipulators, we introduce a new concept of generalized shoulder-elbow-wrist (SEW) angle, a generalization of the conventional SEW angle but with an arbitrary choice of the reference direction function. The SEW angle is widely used and easy for human operators to visualize as a rotation of the elbow about the shoulder-wrist line. Since other redundancy parameterizations including the conventional SEW angle encounter an algorithmic singularity along a line in the workspace, we introduce a special choice of the reference direction function called the stereographic SEW angle which has a singularity only along a half-line, which can be placed out of reach. We prove that such a singularity is unavoidable for any parameterization. We also include expressions for the SEW angle Jacobian along with singularity analysis. Finally, we provide efficient and singularity-robust inverse kinematics solutions for most known 7R manipulators using the general SEW angle and the subproblem decomposition method. These solutions are often closed-form but may sometimes involve a 1D or 2D search in the general case. Search-based solutions may be converted to finding zeros of a high-order polynomial. Inverse kinematics solutions, examples, and evaluations are available in a publicly accessible repository.

Figures (14)

• Figure 1: Self-motion for a Motoman SIA50D (R-R-3RE-2R) robot arm. For a given end effector pose, the spherical elbow may be on a curve which is the intersection of a torus formed by joints 1 and 2 a sphere centered at joints 6 and 7.
• Figure 2: General 7R robot arm with example shoulder, elbow, and wrist points ${\mathcal{O}}_S$, ${\mathcal{O}}_E$, and ${\mathcal{O}}_W$ fixed in ${\mathcal{E}}_0$, ${\mathcal{E}}_3$, and ${\mathcal{E}}_7$, respectively. In general, these points may be placed anywhere in the kinematic chain.
• Figure 3: A 3R|| joint is the limit of a 3R joint as the intersection point moves to infinity. If this joint is the elbow, then although in the limit $p_{SE}$ has infinite length, the normalized vector $e_{SE}$ is defined and is equal to the three parallel joint axes.
• Figure 4: The SEW angle $\psi$ is the angle of the elbow measured from $e_x=f_x(p_{SW})$ about $e_{SW}$, which is also the angle of the SEW plane normal vector $n_{SEW}$ measured from $e_y$ about $e_{SW}$. In the general SEW angle, the reference direction function $f_x(p_{SW})$ is arbitrary but with the constraints that the output is unit length and orthogonal to $p_{SW}$.
• Figure 5: Geometric interpretations for $e_x = f_x(p_{SW})$. (a) In the conventional SEW angle, $e_x$ is the normalized version of the component of $e_r$ orthogonal to $e_{SW}$, and $e_y$ is normal to the plane containing $e_{SW}$ and $e_r$. (b) In the stereographic SEW angle, $k_{rt}$ is normal to the plane containing $e_{SW}-e_t$ and $e_r$, $e_y$ is the normalized version of the component of $k_{rt}$ orthogonal to $e_{SW}$, and $e_x$ is normal to $k_{rt}$ and $e_{SW}$.