• Table of Contents

Introduction 👋

As medical robots continue to offer unprecedented precision and consistency in challenging procedures, understanding the kinematic foundations of these systems becomes crucial for optimizing their medical applications. This report starts off with a detailed kinematic analysis of the much simpler 4DOF Trossen PincherX 100 robotic arm, examining its joint parameters and link variables. Through the derivation of the Denavit-Hartenberg parameters and transformation matrices for each link, I establish a simulation framework for analyzing the robot's workspace and kinematic limitations. In the end, I contrast these findings with the CyberKnife®, a much more sophisticated 6DOF medical robot specifically designed for radiation therapy. By examining how kinematic configurations for different robots affect performance in the clinical setting, I demonstrate the critical relationship between degrees of freedom, workspace characteristics and medical efficacy.

Throughout this report, I will be using the custom simulations and robot models I’ve built. If you'd like to experiment with the interactive demos, I recommend cloning my code repository and downloading the modified model shared on Fusion Team [3].

Engineering drawing by Trossen Robotics [1]

PincherX-100 Solid STEP Files by Interbotix [1]

For reference, all simulations and model files are built using the engineering drawing shown on the left. Moreover, Interbotix has provided detailed specifications—including joint limits and the original model STEP files—here. I have incorporated the joints and their joint limits from the table below into the Interbotix model using Autodesk Fusion. As noted above, you can explore the arm's full range of motion with my publicly available model on Fusion Team here [3].

Default joint limits provided by Interbotix [1]

The PincherX 100 model with fully functional joints and joint limits [3]

PincherX 100 🦾

The Trossen PincherX 100 consists of 5 links, as shown in the Fusion model provided by the Interbotix Arms Documentation [1]. The robot's structure is composed of the following joints and links:

• Waist: The base of the robot (link 0), featuring a revolute joint that rotates around the z-axis.
• Shoulder: The second link (link 1), connected to the waist via a revolute joint.
• Elbow: The third link (link 2), connected to the shoulder via a revolute joint.
• Forearm: The fourth link (link 3), connected to the elbow via a revolute joint.
• Wrist: The final link (link 4) that attaches to the end effector.

Waist (base + revolute joint 1) [1]

Elbow (link 2 + revolute joint 3) [1]

Wrist (link 4) [1]

Shoulder (link 1 + revolute joint 2) [1]

Forearm (link 3 + revolute joint 4) [1]

Gripper (end effector) [1]

Using Grübler's formula for mobility, we can determine that the Trossen Pincher X Robot has 4 degrees of freedom, as verified on the Interbotix X-Series Arms Documentation.

$$ M = 6(L-N-1) + \sum_{j} f_j $$

$$ M = 6(5-4-1) +4(1) $$

$$ M = 4 $$

• M = Mobility (degrees of freedom)
• L = Number of links (including base/ground)
• N = Number of joints
• fⱼ = Degrees of freedom for each joint

The formula multiplies by 6 to account for the degrees of freedom of a rigid body in space.

Link Parameters and Joint Variables

Identifying the link parameters and joint variables is relatively straightforward after figuring out the links and joints. Recall that the Denavit-Hartenberg (DH) parameters consist of four values for each link: