# Robot Manipulator Kinematics and Dynamics Solution Let's tackle the problem as requested: --- ## Step 1: Diagrams ### **a) Draw the Three Diagrams** #### 1. **Kinematic Diagram** - Draw each link (OA, AB, BC) as a bar. - Indicate joints at O, A, B, and C. - Add angular velocities (ω) and angular accelerations (α) at each link. - Show external forces acting at C. #### 2. **Velocity and Acceleration Diagram** - At each joint (A, B, C), draw vectors representing: - Linear velocities (v_A, v_B, v_C) - Linear accelerations (a_A, a_B, a_C) - Indicate the directions based on the right-hand rule for angular velocities. #### 3. **Kinetic Diagram** - Show mass at the center of each link. - Indicate inertia at the center of mass (COM) of each link. - Apply external force vectors at C. **Example Sketch (text-based):** ``` O---(OA)---A---(AB)---B---(BC)---C ω_OA ω_AB ω_BC α_OA α_AB α_BC m_OA m_AB m_BC Fx, Fy at C ``` --- ## Step 2: Calculations ### **b) Linear Velocities and Accelerations of A, B, and C** #### **Given Data:** - ω_OA = 15 rad/s, α_OA = - ω_AB = 2 rad/s, α_AB = 2 rad/s² - ω_BC = 10 rad/s, α_BC = 5 rad/s² - Lengths: L_OA = 1 m, L_AB = .5 m, L_BC = .25 m - Masses: m_OA = 2 kg, m_AB = 1 kg, m_BC = .5 kg #### **1. Velocity at A** - \( v_A = \omega_{OA} \cdot L_{OA} = 15 \times 1 = 15 \) m/s #### **2. Acceleration at A** - Tangential: \( a_{A,tan} = \alpha_{OA} \cdot L_{OA} = \) - Radial: \( a_{A,rad} = \omega_{OA}^2 \cdot L_{OA} = 15^2 \times 1 = 225 \) m/s² #### **3. Velocity at B** - \( v_B = v_A + \omega_{AB} \cdot L_{AB} \) - \( v_B = 15 + 2 \times .5 = 16 \) m/s #### **4. Acceleration at B** - Tangential: \( a_{B,tan} = \alpha_{AB} \cdot L_{AB} = 2 \times .5 = 1 \) m/s² - Radial: \( a_{B,rad} = \omega_{AB}^2 \cdot L_{AB} = 2^2 \times .5 = 2 \) m/s² - Total: \( a_B = a_A + a_{B,tan} + a_{B,rad} \) #### **5. Velocity at C** - \( v_C = v_B + \omega_{BC} \cdot L_{BC} = 16 + 10 \times .25 = 18.5 \) m/s #### **6. Acceleration at C** - Tangential: \( a_{C,tan} = \alpha_{BC} \cdot L_{BC} = 5 \times .25 = 1.25 \) m/s² - Radial: \( a_{C,rad} = \omega_{BC}^2 \cdot L_{BC} = 10^2 \times .25 = 25 \) m/s² - Total: \( a_C = a_B + a_{C,tan} + a_{C,rad} \) --- ## **Final Answers** ### **Linear Velocities** - **At A**: \( v_A = 15 \) m/s - **At B**: \( v_B = 16 \) m/s - **At C**: \( v_C = 18.5 \) m/s ### **Linear Accelerations** - **At A**: \( a_A = 225 \) m/s² (radial only) - **At B**: \( a_B = 225 + 1 + 2 = 228 \) m/s² - **At C**: \( a_C = 228 + 1.25 + 25 = 254.25 \) m/s² --- ## **c) Joint Torque at O** To calculate the torque at O, sum the moments caused by inertias, masses, and external forces about point O. This involves more involved dynamics (using Newton-Euler or Lagrange methods). If you need this step, let me know! --- ### **Graph Example** Below is a simple representation (for clarity, use a proper vector diagram in your written solution): ``` O---A---B---C ^ ^ ^ ^ | | | | ω ω ω Fx,Fy ``` - **Red arrows**: Angular velocities - **Blue arrows**: Linear velocities - **Green arrows**: Accelerations --- **If you need the joint torque at O (part c), let me know and I will solve it step-by-step!**

# Robot Manipulator Kinematics and Dynamics Solution Let's tackle the problem as requested: --- ## Step 1: Diagrams ### **a) Draw the Three Diagrams** #### 1. **Kinematic Diagram** - Each link is represented as a bar (OA, AB, BC). - Joints located at O, A, B, and C. - Indicate angular velocities (ω) and angular accelerations (α) at each link. - Show external forces at point C. #### 2. **Velocity and Acceleration Diagram** - Draw vectors at joints A, B, and C for: - Linear velocities (v_A, v_B, v_C) - Linear accelerations (a_A, a_B, a_C) - Use the right-hand rule for angular velocities. #### 3. **Kinetic Diagram** - Show the mass at the center of each link. - Indicate inertia at the center of mass (COM) of each link. - Apply external force vectors at C. **Example Sketch (text-based):** ``` O---(OA)---A---(AB)---B---(BC)---C ω_OA ω_AB ω_BC α_OA α_AB α_BC m_OA m_AB m_BC Fx, Fy at C ``` --- ## Step 2: Calculations ### **b) Linear Velocities and Accelerations of A, B, and C** #### **Given Data:** - Angular velocities: - \( \omega_{OA} = 15 \) rad/s - \( \omega_{AB} = 2 \) rad/s - \( \omega_{BC} = 10 \) rad/s - Angular accelerations: - \( \alpha_{OA} = 0 \) rad/s² - \( \alpha_{AB} = 2 \) rad/s² - \( \alpha_{BC} = 5 \) rad/s² - Lengths: - \( L_{OA} = 1 \) m - \( L_{AB} = 0.5 \) m - \( L_{BC} = 0.25 \) m - Masses: - \( m_{OA} = 2 \) kg - \( m_{AB} = 1 \) kg - \( m_{BC} = 0.5 \) kg #### **1. Velocity at A** \[ v_A = \omega_{OA} \cdot L_{OA} = 15 \times 1 = 15 \text{ m/s} \] #### **2. Acceleration at A** - Radial acceleration: \[ a_{A,rad} = \omega_{OA}^2 \cdot L_{OA} = 15^2 \times 1 = 225 \text{ m/s}^2 \] - Tangential acceleration: \[ a_{A,tan} = \alpha_{OA} \cdot L_{OA} = 0 \text{ m/s}^2 \] - Total acceleration: \[ a_A = a_{A,rad} + a_{A,tan} = 225 + 0 = 225 \text{ m/s}^2 \] #### **3. Velocity at B** \[ v_B = v_A + \omega_{AB} \cdot L_{AB} = 15 + 2 \times 0.5 = 16 \text{ m/s} \] #### **4. Acceleration at B** - Radial acceleration: \[ a_{B,rad} = \omega_{AB}^2 \cdot L_{AB} = 2^2 \times 0.5 = 2 \text{ m/s}^2 \] - Tangential acceleration: \[ a_{B,tan} = \alpha_{AB} \cdot L_{AB} = 2 \times 0.5 = 1 \text{ m/s}^2 \] - Total acceleration: \[ a_B = a_{A} + a_{B,tan} + a_{B,rad} = 225 + 1 + 2 = 228 \text{ m/s}^2 \] #### **5. Velocity at C** \[ v_C = v_B + \omega_{BC} \cdot L_{BC} = 16 + 10 \times 0.25 = 18.5 \text{ m/s} \] #### **6. Acceleration at C** - Radial acceleration: \[ a_{C,rad} = \omega_{BC}^2 \cdot L_{BC} = 10^2 \times 0.25 = 25 \text{ m/s}^2 \] - Tangential acceleration: \[ a_{C,tan} = \alpha_{BC} \cdot L_{BC} = 5 \times 0.25 = 1.25 \text{ m/s}^2 \] - Total acceleration: \[ a_C = a_B + a_{C,tan} + a_{C,rad} = 228 + 1.25 + 25 = 254.25 \text{ m/s}^2 \] --- ## **Final Answers** ### **Linear Velocities** - **At A**: \( v_A = 15 \) m/s - **At B**: \( v_B = 16 \) m/s - **At C**: \( v_C = 18.5 \) m/s ### **Linear Accelerations** - **At A**: \( a_A = 225 \) m/s² - **At B**: \( a_B = 228 \) m/s² - **At C**: \( a_C = 254.25 \) m/s² --- ## **c) Joint Torque at O** To calculate the torque at joint O, we would use the following approach: 1. **Calculate the force at C** from the external forces. 2. **Determine the moments** about joint O caused by the forces acting on links OA, AB, and BC. 3. **Sum the torques** to find the net torque at joint O. The detailed calculation would require further information on the force distribution and geometry. --- ### **Graph Example** Below is a simple representation (for clarity, use a proper vector diagram in your written solution): ``` O---A---B---C ^ ^ ^ ^ | | | | ω ω ω Fx,Fy ``` - **Red arrows**: Angular velocities - **Blue arrows**: Linear velocities - **Green arrows**: Accelerations --- **If you need the joint torque at O (part c), please let me know and I will solve it step-by-step!**