Let's break down the problem **step by step**: --- ### **Step 1: Equations for Revolute Joint (Given)** For a revolute joint \(i\): - **Linear velocity contribution:** \[ {^0}\mathbf{v}_{i} = \dot{\theta}_i \; {^0}\mathbf{Z}_i \times ({^0}\mathbf{P}_{tool} - {^0}\mathbf{P}_{i_{org}}) \] (Eq. 5.110) - **Angular velocity contribution:** \[ {^0}\boldsymbol{\omega}_{i} = \dot{\theta}_i \; {^0}\mathbf{Z}_i \] (Eq. 5.111) --- ### **Step 2: Analogous Equations for Prismatic Joint** For a prismatic joint \(i\): - **Linear velocity contribution:** For a prismatic joint, the joint variable is the displacement \(d_i\), and the motion is along the \(Z_i\) direction: \[ {^0}\mathbf{v}_{i} = \dot{d}_i \; {^0}\mathbf{Z}_i \] - **Angular velocity contribution:** Prismatic joints do **not** contribute to angular velocity: \[ {^0}\boldsymbol{\omega}_{i} = \mathbf{0} \] --- ### **Step 3: Write the Jacobian Matrix for 6-DOF Manipulator** The Jacobian relates joint velocities to the end-effector's linear and angular velocities: \[ \begin{bmatrix} \mathbf{v}_{tool} \\ \boldsymbol{\omega}_{tool} \end{bmatrix} = \mathbf{J} \; \dot{\mathbf{q}} \] Each column \(i\) of the Jacobian corresponds to joint \(i\): #### **If joint \(i\) is revolute:** \[ \mathbf{J}_{v,i} = {^0}\mathbf{Z}_i \times ({^0}\mathbf{P}_{tool} - {^0}\mathbf{P}_{i_{org}}) \] \[ \mathbf{J}_{\omega,i} = {^0}\mathbf{Z}_i \] #### **If joint \(i\) is prismatic:** \[ \mathbf{J}_{v,i} = {^0}\mathbf{Z}_i \] \[ \mathbf{J}_{\omega,i} = \mathbf{0} \] --- ### **Step 4: Write the General Jacobian Matrix** So, the **general \(6 \times 6\) Jacobian** for a 6-DOF manipulator can be written as: \[ \mathbf{J}_i = \begin{cases} \begin{bmatrix} {^0}\mathbf{Z}_i \times ({^0}\mathbf{P}_{tool} - {^0}\mathbf{P}_{i_{org}}) \\ {^0}\mathbf{Z}_i \end{bmatrix} & \text{if joint \(i\) is revolute} \\ \\ \begin{bmatrix} {^0}\mathbf{Z}_i \\ \mathbf{0} \end{bmatrix} & \text{if joint \(i\) is prismatic} \end{cases} \] Or, **for all joints:** \[ \mathbf{J} = \begin{bmatrix} \mathbf{J}_{v,1} & \mathbf{J}_{v,2} & \cdots & \mathbf{J}_{v,6} \\ \mathbf{J}_{\omega,1} & \mathbf{J}_{\omega,2} & \cdots & \mathbf{J}_{\omega,6} \end{bmatrix} \] where \[ \mathbf{J}_{v,i} = \begin{cases} {^0}\mathbf{Z}_i \times ({^0}\mathbf{P}_{tool} - {^0}\mathbf{P}_{i_{org}}), & \text{revolute} \\ {^0}\mathbf{Z}_i, & \text{prismatic} \end{cases} \] \[ \mathbf{J}_{\omega,i} = \begin{cases} {^0}\mathbf{Z}_i, & \text{revolute} \\ \mathbf{0}, & \text{prismatic} \end{cases} \] --- ### **Final Answer (Summary Table)** | Joint Type | \(\mathbf{J}_{v,i}\) | \(\mathbf{J}_{\omega,i}\) | |---------------|------------------------------------------------------------------|-------------------------------------| | Revolute | \({^0}\mathbf{Z}_i \times ({^0}\mathbf{P}_{tool} - {^0}\mathbf{P}_{i_{org}})\) | \({^0}\mathbf{Z}_i\) | | Prismatic | \({^0}\mathbf{Z}_i\) | \(\mathbf{0}\) | Where: - \({^0}\mathbf{Z}_i\): Unit Z vector of frame \(i\) in base frame - \({^0}\mathbf{P}_{i_{org}}\): Origin of frame \(i\) in base frame - \({^0}\mathbf{P}_{tool}\): Tool point in base frame --- **This is the step-by-step solution and the final form of the 6x6 Jacobian in terms of the given quantities.**

Certainly! Let's walk through the detailed step-by-step derivation, including explanations, reasoning, and calculations. --- ## **Problem Restatement** Given a 6-DOF robot manipulator: - For each link \(i\), we know: - The unit vector \(\mathbf{Z}_i\) (direction of joint axis) in the base frame. - The position vector \(\mathbf{P}_i\) (origin of link \(i\)) in the base frame. - The goal: - To find the velocity of the tool point (fixed relative to link \(n\)) in terms of the joint velocities. - To derive equations analogous to (5.110) and (5.111) for **prismatic joints**. - To formulate the **6×6 Jacobian matrix** in terms of \(\mathbf{Z}_i\), \(\mathbf{P}_i\), and the position of the tool point \(\mathbf{P}_{tool}\). --- ## **Step 1: Understanding the given equations for revolute joints** ### **Revolute Joint** - **Angular velocity contribution:** \[ \boxed{ {^0}\boldsymbol{\omega}_i = \dot{\theta}_i \; {^0}\mathbf{Z}_i } \] - **Linear velocity contribution:** \[ \boxed{ {^0}\mathbf{v}_i = \dot{\theta}_i \; {^0}\mathbf{Z}_i \times \left( {^0}\mathbf{P}_{tool} - {^0}\mathbf{P}_i \right) } \] **Interpretation:** - Rotation about axis \(\mathbf{Z}_i\) with rate \(\dot{\theta}_i\). - The linear velocity at the tool point due to joint \(i\) is the cross product of \(\mathbf{Z}_i\) and the vector from joint \(i\) to the tool point. --- ## **Step 2: Equations for prismatic joints** ### **Prismatic Joint** - **Linear velocity:** Since prismatic joints translate along their axis: \[ {^0}\mathbf{v}_i = \dot{d}_i \; {^0}\mathbf{Z}_i \] - **Angular velocity:** Prismatic joints do **not** contribute to angular velocity: \[ {^0}\boldsymbol{\omega}_i = \mathbf{0} \] **Summary:** | Joint Type | Linear velocity contribution \(\mathbf{v}_i\) | Angular velocity contribution \(\boldsymbol{\omega}_i\) | |--------------|--------------------------------------------------|--------------------------------------------------------| | Revolute | \(\dot{\theta}_i \; {^0}\mathbf{Z}_i \times \left( {^0}\mathbf{P}_{tool} - {^0}\mathbf{P}_i \right)\) | \(\dot{\theta}_i \; {^0}\mathbf{Z}_i\) | | Prismatic | \(\dot{d}_i \; {^0}\mathbf{Z}_i\) | \(\mathbf{0}\) | --- ## **Step 3: Expressing the Jacobian** The **Jacobian** relates joint velocities \(\dot{\mathbf{q}}\) (vector of all joint rates) to the end-effector (tool point) velocities: \[ \boxed{ \begin{bmatrix} {^0}\mathbf{v}_{tool} \\ {^0}\boldsymbol{\omega}_{tool} \end{bmatrix} = \mathbf{J} \; \dot{\mathbf{q}} } \] where: - \(\mathbf{J}\) is a \(6 \times 6\) matrix composed of columns \(\mathbf{J}_i\), each associated with joint \(i\). --- ## **Step 4: Derive the Jacobian columns for each joint** ### **For joint \(i\), the contribution is:** \[ \mathbf{J}_i = \begin{bmatrix} \mathbf{J}_{v,i} \\ \mathbf{J}_{\omega,i} \end{bmatrix} \] - **Revolute joint:** \[ \boxed{ \mathbf{J}_{v,i} = {^0}\mathbf{Z}_i \times \left( {^0}\mathbf{P}_{tool} - {^0}\mathbf{P}_i \right) } \] \[ \boxed{ \mathbf{J}_{\omega,i} = {^0}\mathbf{Z}_i } \] - **Prismatic joint:** \[ \boxed{ \mathbf{J}_{v,i} = {^0}\mathbf{Z}_i } \] \[ \boxed{ \mathbf{J}_{\omega,i} = \mathbf{0} } \] --- ## **Step 5: Construct the full Jacobian matrix** Putting it all together: \[ \boxed{ \mathbf{J} = \begin{bmatrix} \mathbf{J}_{v,1} & \mathbf{J}_{v,2} & \dots & \mathbf{J}_{v,6} \\ \mathbf{J}_{\omega,1} & \mathbf{J}_{\omega,2} & \dots & \mathbf{J}_{\omega,6} \end{bmatrix} } \] with each \(\mathbf{J}_{v,i}\) and \(\mathbf{J}_{\omega,i}\) as above. --- ## **Final Formulation:** ### **For each joint \(i\):** \[ \boxed{ \mathbf{J}_i = \begin{bmatrix} \mathbf{J}_{v,i} \\ \mathbf{J}_{\omega,i} \end{bmatrix} = \begin{cases} \begin{bmatrix} {^0}\mathbf{Z}_i \times \left( {^0}\mathbf{P}_{tool} - {^0}\mathbf{P}_i \right) \\ {^0}\mathbf{Z}_i \end{bmatrix} & \text{if joint \(i\) is revolute} \\ \\ \begin{bmatrix} {^0}\mathbf{Z}_i \\ \mathbf{0} \end{bmatrix} & \text{if joint \(i\) is prismatic} \end{cases} } \] --- ## **Summary of Key Equations:** | **Joint Type** | **Linear velocity component \(\mathbf{J}_{v,i}\)** | **Angular velocity component \(\mathbf{J}_{\omega,i}\)** | |----------------|--------------------------------------------------------|---------------------------------------------------------| | **Revolute** | \(\boxed{{^0}\mathbf{Z}_i \times \left( {^0}\mathbf{P}_{tool} - {^0}\mathbf{P}_i \right)}\) | \(\boxed{{^0}\mathbf{Z}_i}\) | | **Prismatic** | \(\boxed{{^0}\mathbf{Z}_i}\) | \(\boxed{\mathbf{0}}\) | --- ## **Note:** - \({^0}\mathbf{Z}_i\) is the unit vector of joint \(i\)’s axis in the base frame. - \({^0}\mathbf{P}_i\) is the position of joint \(i\)’s origin in the base frame. - \({^0}\mathbf{P}_{tool}\) is the position of the tool point in the base frame. --- ## **Final statement:** **The 6x6 Jacobian matrix** for a general 6-DOF manipulator, considering both revolute and prismatic joints, is: \[ \boxed{ \mathbf{J} = \begin{bmatrix} \mathbf{J}_v \\ \mathbf{J}_\omega \end{bmatrix} \] where \[ \mathbf{J}_v = \left[ \mathbf{J}_{v,1} \quad \mathbf{J}_{v,2} \quad \dots \quad \mathbf{J}_{v,6} \right], \quad \mathbf{J}_\omega = \left[ \mathbf{J}_{\omega,1} \quad \mathbf{J}_{\omega,2} \quad \dots \quad \mathbf{J}_{\omega,6} \right] \] with each \(\mathbf{J}_{v,i}\), \(\mathbf{J}_{\omega,i}\) as above. --- This detailed derivation provides the **step-by-step reasoning**, **equations**, and **final form** of the Jacobian for any 6-DOF robot with mixed joint types.