Chapter 6 • Theory & Derivations

Perception, Navigation and Decision-Making

Comprehensive university textbook unit covering fundamental principles, governing relationships, and engineering applications of Perception, Navigation and Decision-Making.

§6.1 Cameras, depth sensors and calibration

**OpenSTEM Mechanical Engineering · Book 12: Robotics and Automation · Unit 6: Perception, Navigation and Decision-Making.** Educational draft. All worked numbers are illustrative unless stated otherwise; do not use the calculations alone to certify a physical robot or protective system.

§6.2 Detection confidence and classification

§6.3 Graph and grid path planning

§6.4 Reactive obstacle avoidance

§6.5 Task planning and finite-state execution

THREE WORKED QUESTIONS

Step-by-Step Robotics and Automation Solutions

Three original questions connect calculations, definitions and interpretation. Open each solution to follow the reasoning.

Solved Problem Example 6.1: Calculation & Verification: Cameras, depth sensors and calibration

Problem Statement: An engineering practitioner analyzes a mechanical system governed by **Cameras, depth sensors and calibration** under steady operational loading.

Worked Engineering Derivation:

For f=500 pixel, baseline=0.10 m and disparity=50 pixel, depth is 1.0 m in the ideal model. Reproduce the substitution on paper, carry units through each stage and check the scale with an independent estimate. Precision in displayed decimal places does not demonstrate precise knowledge of link lengths, accelerations, friction coefficients or camera calibration. If inputs are rounded or synthetic, the outputs inherit those limitations; do not label them as measured robot performance. **Discretization audit.** Choose a simulation step that resolves the fastest physical phenomenon of interest. Check convergence with a smaller step and confirm that numerical artifacts do not explain apparent stability or contact behavior. Sampling rates must exceed the bandwidth of the phenomena they are claimed to detect. Extend the example by sweeping one independent input through a realistic range. Predict the qualitative trend before computing. Then change another input in a separate run; explain why changing both simultaneously would make causal interpretation weaker. Document any singular, discontinuous or saturating regime instead of hiding it by clipping a plot without explanation.

Solved Problem Example 6.2: Calculation & Verification: Detection confidence and classification

Problem Statement: An engineering practitioner analyzes a mechanical system governed by **Detection confidence and classification** under steady operational loading.

Worked Engineering Derivation:

With 80 true positives and 20 false positives, sample precision is 0.80; this does not measure recall without false negatives. Reproduce the substitution on paper, carry units through each stage and check the scale with an independent estimate. Precision in displayed decimal places does not demonstrate precise knowledge of link lengths, accelerations, friction coefficients or camera calibration. If inputs are rounded or synthetic, the outputs inherit those limitations; do not label them as measured robot performance. **Uncertainty audit.** Differentiate precision from accuracy: repeated values near one another may share a systematic offset. Use replicated trials and a calibration reference, and consider correlations when combining uncertainties. An error bound must be linked to a defined operating range. Extend the example by sweeping one independent input through a realistic range. Predict the qualitative trend before computing. Then change another input in a separate run; explain why changing both simultaneously would make causal interpretation weaker. Document any singular, discontinuous or saturating regime instead of hiding it by clipping a plot without explanation.

Solved Problem Example 6.3: Calculation & Verification: Graph and grid path planning

Problem Statement: For relative cell offsets dx=4, dy=3, Manhattan lower bound is seven unit moves when diagonal actions are disallowed. Reproduce the substitution on paper, carry units through each stage and check the scale with an independent estimate. Precision in displayed decimal places does not demonstrate precise knowledge of link lengths, accelerations, friction coefficients or camera calibration. If inputs are rounded or synthetic, the outputs inherit those limitations; do not label them as measured robot performance.

Worked Engineering Derivation:

**Traceable Verification audit.** Record firmware or model version, sensor calibration, test fixture revision and all rejected trials. Set acceptance thresholds before viewing the final traces. If a physical trial cannot be carried out safely, use simulation only and write the missing validation item explicitly. Extend the example by sweeping one independent input through a realistic range. Predict the qualitative trend before computing. Then change another input in a separate run; explain why changing both simultaneously would make causal interpretation weaker. Document any singular, discontinuous or saturating regime instead of hiding it by clipping a plot without explanation.

Solved Problem Example 6.4: Calculation & Verification: Reactive obstacle avoidance

Problem Statement: An engineering practitioner analyzes a mechanical system governed by **Reactive obstacle avoidance** under steady operational loading.

Worked Engineering Derivation:

At speed 0.80 m/s, delay 0.10 s and deceleration 1.0 m/s², ideal stop distance is 0.40 m before uncertainties. Reproduce the substitution on paper, carry units through each stage and check the scale with an independent estimate. Precision in displayed decimal places does not demonstrate precise knowledge of link lengths, accelerations, friction coefficients or camera calibration. If inputs are rounded or synthetic, the outputs inherit those limitations; do not label them as measured robot performance. **Dimensional Balance audit.** Perform a units check by substituting SI base units and showing both sides match. For angles, state that trigonometric functions expect radians unless explicitly converted. For probabilities, check output within [0,1], and for lengths or times reject negative values when the model defines only positive quantities. Extend the example by sweeping one independent input through a realistic range. Predict the qualitative trend before computing. Then change another input in a separate run; explain why changing both simultaneously would make causal interpretation weaker. Document any singular, discontinuous or saturating regime instead of hiding it by clipping a plot without explanation.

Solved Problem Example 6.5: Calculation & Verification: Task planning and finite-state execution

Problem Statement: With per-attempt success probability 0.80, expected attempts are 1.25 only if trials are independent and identical. Reproduce the substitution on paper, carry units through each stage and check the scale with an independent estimate. Precision in displayed decimal places does not demonstrate precise knowledge of link lengths, accelerations, friction coefficients or camera calibration. If inputs are rounded or synthetic, the outputs inherit those limitations; do not label them as measured robot performance.

Worked Engineering Derivation:

**Boundary Cases audit.** Check a zero-input limit, a typical operating point and a maximum planned input. Where derivatives exist, estimate local sensitivity to a 5% change. If the result changes sharply, prioritize a measurement of the sensitive parameter rather than increasing numerical precision elsewhere. Extend the example by sweeping one independent input through a realistic range. Predict the qualitative trend before computing. Then change another input in a separate run; explain why changing both simultaneously would make causal interpretation weaker. Document any singular, discontinuous or saturating regime instead of hiding it by clipping a plot without explanation.