Robotics, Kinematics and Motion Planning
Comprehensive university textbook unit covering fundamental principles, governing relationships, and engineering applications of Robotics, Kinematics and Motion Planning.
§6.1 Two-link forward kinematics
**OpenSTEM Mechanical Engineering · Book 11: Mechatronics Engineering · Unit 6: Robotics, Kinematics and Motion Planning.** Educational draft for university study; no hardware-specific safety certification is implied.
§6.2 Inverse kinematics and multiple branches
**OpenSTEM Mechanical Engineering · Book 11: Mechatronics Engineering · Unit 6: Robotics, Kinematics and Motion Planning.** Educational draft for university study; no hardware-specific safety certification is implied.
§6.3 Smooth trajectories and jerk tradeoffs
**OpenSTEM Mechanical Engineering · Book 11: Mechatronics Engineering · Unit 6: Robotics, Kinematics and Motion Planning.** Educational draft for university study; no hardware-specific safety certification is implied.
§6.4 Backlash, compliance and end-effector accuracy
**OpenSTEM Mechanical Engineering · Book 11: Mechatronics Engineering · Unit 6: Robotics, Kinematics and Motion Planning.** Educational draft for university study; no hardware-specific safety certification is implied.
§6.5 Workspace boundaries and collision prevention
**OpenSTEM Mechanical Engineering · Book 11: Mechatronics Engineering · Unit 6: Robotics, Kinematics and Motion Planning.** Educational draft for university study; no hardware-specific safety certification is implied.
Step-by-Step Mechatronics Engineering Solutions
Three original questions connect calculations, definitions and interpretation. Open each solution to follow the reasoning.
Problem Statement: For L1=0.30 m, L2=0.20 m, q1=0 and q2=90 degrees, the tip is at (0.30,0.20) m. Rewrite the given numbers before substitution; explicitly carry their units through the equation. Compare the answer to a rough mental estimate and state a sensible number of significant figures. Do not interpret the displayed precision as measured accuracy. The calculation is synthetic unless independently verified against a particular device.
Worked Engineering Derivation:
**Sensitivity experiment.** Repeat the calculation after changing one controlling quantity by +10% while keeping other stated inputs fixed. For a linear relationship, expect a proportional response in the appropriate variable; for a nonlinear one, compute the actual change and compare it with the local slope. Explain whether the chosen perturbation is realistic for the application. **Dimensional audit.** Convert angles, voltages, forces, lengths and time to an explicitly declared convention before putting them in code. A numerical answer with the wrong units is not a small editorial error: it can change the actual machine motion or power demands by large factors. Record the conversion and the numerical rounding separately.
Problem Statement: For L1=0.30 m, L2=0.20 m and target radius 0.40 m, cos(q2)=0.25 and q2≈75.52 degrees in one branch. Rewrite the given numbers before substitution; explicitly carry their units through the equation. Compare the answer to a rough mental estimate and state a sensible number of significant figures. Do not interpret the displayed precision as measured accuracy. The calculation is synthetic unless independently verified against a particular device.
Worked Engineering Derivation:
**Sensitivity experiment.** Repeat the calculation after changing one controlling quantity by +10% while keeping other stated inputs fixed. For a linear relationship, expect a proportional response in the appropriate variable; for a nonlinear one, compute the actual change and compare it with the local slope. Explain whether the chosen perturbation is realistic for the application. **Dimensional audit.** Convert angles, voltages, forces, lengths and time to an explicitly declared convention before putting them in code. A numerical answer with the wrong units is not a small editorial error: it can change the actual machine motion or power demands by large factors. Record the conversion and the numerical rounding separately.
Problem Statement: At normalized time u=0.25 a cubic profile has completed 15.625% of the position change. Rewrite the given numbers before substitution; explicitly carry their units through the equation. Compare the answer to a rough mental estimate and state a sensible number of significant figures. Do not interpret the displayed precision as measured accuracy. The calculation is synthetic unless independently verified against a particular device.
Worked Engineering Derivation:
**Sensitivity experiment.** Repeat the calculation after changing one controlling quantity by +10% while keeping other stated inputs fixed. For a linear relationship, expect a proportional response in the appropriate variable; for a nonlinear one, compute the actual change and compare it with the local slope. Explain whether the chosen perturbation is realistic for the application. **Dimensional audit.** Convert angles, voltages, forces, lengths and time to an explicitly declared convention before putting them in code. A numerical answer with the wrong units is not a small editorial error: it can change the actual machine motion or power demands by large factors. Record the conversion and the numerical rounding separately.
Problem Statement: A 60 N lateral load on a 30,000 N/m equivalent stiffness produces 2 mm deflection. Rewrite the given numbers before substitution; explicitly carry their units through the equation. Compare the answer to a rough mental estimate and state a sensible number of significant figures. Do not interpret the displayed precision as measured accuracy. The calculation is synthetic unless independently verified against a particular device.
Worked Engineering Derivation:
**Sensitivity experiment.** Repeat the calculation after changing one controlling quantity by +10% while keeping other stated inputs fixed. For a linear relationship, expect a proportional response in the appropriate variable; for a nonlinear one, compute the actual change and compare it with the local slope. Explain whether the chosen perturbation is realistic for the application. **Dimensional audit.** Convert angles, voltages, forces, lengths and time to an explicitly declared convention before putting them in code. A numerical answer with the wrong units is not a small editorial error: it can change the actual machine motion or power demands by large factors. Record the conversion and the numerical rounding separately.
Problem Statement: At 0.50 m/s and effective deceleration 2.0 m/s^2, idealized stopping travel is at least 0.0625 m before delays. Rewrite the given numbers before substitution; explicitly carry their units through the equation. Compare the answer to a rough mental estimate and state a sensible number of significant figures. Do not interpret the displayed precision as measured accuracy. The calculation is synthetic unless independently verified against a particular device.
Worked Engineering Derivation:
**Sensitivity experiment.** Repeat the calculation after changing one controlling quantity by +10% while keeping other stated inputs fixed. For a linear relationship, expect a proportional response in the appropriate variable; for a nonlinear one, compute the actual change and compare it with the local slope. Explain whether the chosen perturbation is realistic for the application. **Dimensional audit.** Convert angles, voltages, forces, lengths and time to an explicitly declared convention before putting them in code. A numerical answer with the wrong units is not a small editorial error: it can change the actual machine motion or power demands by large factors. Record the conversion and the numerical rounding separately.