NovelWrist.jl

A Julia package for the kinematic analysis of the $2S\underline{P}U+2RSU+1U$ mechanism that is used as wrist mechanism for the humanoid robot RH5v2, developed at the DFKI Robotics Innovation Center and accepted for

• C. Stoeffler, A. Fernandez, H. Peters, M. Schilling and S. Kumar: Kinematic Analysis of a Novel Humanoid Wrist Parallel Mechanism, Advances in Robot Kinematics 2022 (see ARK 2022).

Maintainers:

• Christoph Stoeffler [email protected]
• Adriano Fernandez [email protected]

Introduction

The here presented 2-DOF mechanism for inclination and tilt possesses two double closed-loop chains and allows increased range of motion compared to a classical $2S\underbar{P}U+1U$

Installation

pkg> add NovelWrist

Documentation

Create a new design

A wrist geometry is defined by the constructor taking keyword arguments related to the geometry above. The wrist mechanism at DFKI Bremen has the following geometry:

julia> using NovelWrist

julia> RH5_wrist = WristGeometry(l = (0.045, 0.045), 
                    	         r = (0.049, 0.049), 
                          	 r_ = (0.049, 0.049),
                          	 h = (0.012, 0.012),
                        	 b = ([0.015, -0.178, -0.034], [-0.015, -0.178, -0.034]),
                          	 c = ([0.015, -0.032, 0.011], [-0.015, -0.032, 0.011]),
                          	 e0 = ([0.027, 0, -0.030], [-0.027, 0, -0.030]),
                          	 n = ([1, 0, 0], [-1, 0, 0]),
                          	 actuator_limits = ((0.113, 0.178), (0.113, 0.178))); 

The actuator limits denote the minimum and maximum values that can be reached by the linear actuators, denoted as q in the kinematic model. Presented function calls are executed for the assembly mode of RH5_wrist (solution = [1,2]) but can be altered. Note that the normal vector n has to point outwards on both sides of the mechanism.

Kinematics

Inverse Kinematics

Computation of the actuator length from a given pose defined by inclination ($\alpha$) and tilt ($\gamma$). Note that 'solution' defines which intersection points to pick from both sides of the circle-sphere intersectio. All functions consider intrinsic rotation of the end-effector but it can be changed via intrinsic = false.

julia> x = [0, 0]; # angles in rad  

julia> q = inverse_kinematics(x, RH5_wrist; solution = [1,2])
2-element Vector{Real}:
 0.13347357815533836
 0.13347357815533836

Forward Kinematics

Computes the end-effector orientation α and γ, given the solution for the actuator lengths q:

julia> α, γ = forward_kinematics(q, RH5_wrist, solution = [2,1,1]) 
(-2.2204460492503136e-16, 0.0)

Constrained Jacobian

To get the Jacobian $\mathbf{J}$ as product of the inverted work space Jacobian $\mathbf{J}_x$ and the joint space Jacobian $\mathbf{J}_q$:

julia> J = Jacobian(x, RH5_wrist; specsol = [1,2] split = false)
2×2 Matrix{Real}:
 15.9633   15.9633
 17.737   -17.737

When split = true, $\mathbf{J}_x$ and $\mathbf{J}_q$ are returned componentwise.

Performance Analysis

Conditioning

The condition index of the novel mechanism can be plotted over α and γ:

julia> plot_conditioning(RH5_wrist, α = (-π, π), γ = (-π, π), solution = [1,2], resol = 500) # increasing resol will give a higher resolution

The dashed lines indicate the workspace limits imposed by actuator_limits.

Configuration Space

The actuator lengths for plotting the the configuration space are computed for end-effector orientations between -π and π:

julia> plot_configuration_space(RH5_wrist; solution = [1,2], intrinsic = true, resol = 100)

Here, for better visibility, the actuator_limits are visualized using a red rectangle.

Comparison to Conventional Wrist Designs

Computes and plots the difference of the condition index between $2S\underline{P}U+2RSU+1U$ and $2S\underline{P}U+1U$ mechanism (positive values indicate increased dexterity of the novel design):

julia> plot_comparative_conditioning(RH5_wrist, α = (-π, π), γ = (-π, π), solution = [1,2], resol = 400)

The singularity curves of novel design and comparative design are obtained by sampling through the work space. Note, that in order to get closed contures, a high value for resol has to be set. This however increases the computing time considerably.

julia> plot_comparative_singularities(RH5_wrist, α = (-π, π), γ = (-π, π), solution = [1,2], intrinsic = true, resol = 5000)

The theoretically feasible work space for the novel design is denoted by the blue coloured "shadow".

Plots of Torque and Speed at pure inclination and pure tilt movements can be computed. Additionally, characteristic values are printed to the console:

julia> plot_torque_C(RH5_wrist, α = (-π, π), γ = (-π, π), solution = [1,2], resol=600)
    Pure inclination/tilt characteristics - new wrist:
    Inclination range: -0.74/1.83 rad, 
    Maximum inclination torque: 62.94 Nm, correspondent inclination velocity: 6.36 rad/s, 
    Tilt range: -0.97/0.98 rad, 
    Maximum tilt torque: 56.38 Nm, correspondent tilt velocity: 7.09 rad/s
s
    Pure inclination/tilt characteristics - comparative design:
    Inclination range: -0.74/1.76 rad, 
    Maximum inclination torque: 59.86 Nm, correspondent inclination velocity: 6.68 rad/s, 
    Tilt range: -0.97/0.98 rad, 
    Maximum tilt torque: 53.86 Nm, correspondent tilt velocity: 7.43 rad/s

Acknowledgements

This work was partially supported from the projects VeryHuman (FKZ01IW20004) and TransFIT (FKZ 50RA1701) funded by the German Aerospace Center (DLR) with federal funds from the Federal Ministry of Education and Research (BMBF) and Federal Ministry of Economic Affairs and Energy (BMWi) respectively.