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Robotics models

Connect rigid-body motion, actuators, mobile robots, sensing, and feedback through twenty worked examples.

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1. Forward and inverse kinematics

Definitions & inputs. θ joint angle, l link length, x,y endpoint position; two-link angles are relative joint angles.

  1. Add the two link vectors in the base frame.

    x=l1cos⁡θ1+l2cos⁡(θ1+θ2),y=l1sin⁡θ1+l2sin⁡(θ1+θ2)x=l_1\cos\theta_1+l_2\cos(\theta_1+\theta_2),\quad y=l_1\sin\theta_1+l_2\sin(\theta_1+\theta_2)

  2. Use the cosine rule to invert endpoint distance.

    cos⁡θ2=(x2+y2−l12−l22)/(2l1l2)\cos\theta_2=(x^2+y^2-l_1^2-l_2^2)/(2l_1l_2)

  3. Recover shoulder angle, retaining the chosen elbow branch.

    θ1=atan2⁡(y,x)−atan2⁡(l2sin⁡θ2,l1+l2cos⁡θ2)\theta_1=\operatorname{atan2}(y,x)-\operatorname{atan2}(l_2\sin\theta_2,l_1+l_2\cos\theta_2)

Interpretation. Reachability and multiple solutions must be checked before commanding joints.

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2. Differential motion and forces

Definitions & inputs. q joint coordinates, J endpoint Jacobian, v endpoint velocity, F endpoint force, τ joint torques.

  1. Differentiate forward kinematics.

    x˙=J(q)q˙,Jij=∂xi/∂qj\dot x=J(q)\dot q,\quad J_{ij}=\partial x_i/\partial q_j

  2. Conservation of virtual power maps endpoint force into joint effort.

    FTx˙=τTq˙⇒τ=JTFF^T\dot x=\tau^T\dot q\Rightarrow\tau=J^TF

  3. A straight or folded elbow loses one instantaneous position degree of freedom.

    det⁡J=l1l2sin⁡θ2for planar 2R position\det J= l_1l_2\sin\theta_2\quad\text{for planar 2R position}

Interpretation. Near singularities, small endpoint commands can demand large joint motion.

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3. Mobile motion and actuator dynamics

Definitions & inputs. r wheel radius, b track width, ωR,ωL wheel speeds; Jrot inertia; τ torque.

  1. Average wheel speed translates; the difference rotates.

    v=r(ωR+ωL)/2,ωz=r(ωR−ωL)/bv=r(\omega_R+\omega_L)/2,\quad\omega_z=r(\omega_R-\omega_L)/b

  2. Net torque changes angular acceleration.

    Jrotθ¨=τ−τloadJ_{rot}\ddot\theta=\tau-\tau_{load}

  3. Mechanical power is torque times angular speed.

    P=τω,E=∫PdtP=\tau\omega,\quad E=\int Pdt

Interpretation. Traction, gearbox loss, saturation, and coupling affect real response.

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4. Feedback and uncertainty

Definitions & inputs. e reference minus measurement, Kp proportional gain, R sensor variance, P prior estimate variance.

  1. Proportional action maps tracking error into command.

    u=Kpeu=K_pe

  2. A scalar Kalman update weights an independent measurement by relative uncertainty.

    K=P/(P+R),x^+=x^−+K(z−x^−)K=P/(P+R),\quad\hat x^+=\hat x^-+K(z-\hat x^-)

  3. The posterior variance decreases under the stated independent Gaussian model.

    P+=(1−K)PP^+=(1-K)P

Interpretation. Accurate static calibration does not replace dynamic control validation.

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Graphical worked example

Planar two-link robot with l1=l2=0.5 m; zero determinant marks a position singularity. X axis: Relative elbow angle θ₂ (degrees). Y axis: Jacobian determinant (m²).
Planar two-link robot with l1=l2=0.5 m; zero determinant marks a position singularity. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. One-link endpoint x

Definitions & inputs. l=0.5 m, θ=60°.

  1. Choose the governing model and isolate the requested quantity.

    x=lcos⁡θx=l\cos\theta

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    x=0.5cos⁡60∘x=0.5\cos60^\circ

  3. Evaluate the expression; the result uses the units shown.

    Result=0.25 m\mathrm{Result}=0.25\ {\rm m}

Interpretation. The y coordinate is positive and must be computed separately.

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Example 02. One-link endpoint y

Definitions & inputs. l=0.5 m, θ=60°.

  1. Choose the governing model and isolate the requested quantity.

    y=lsin⁡θy=l\sin\theta

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    y=0.5sin⁡60∘y=0.5\sin60^\circ

  3. Evaluate the expression; the result uses the units shown.

    Result=0.4330127 m\mathrm{Result}=0.4330127\ {\rm m}

Interpretation. Both coordinates refer to the same base frame.

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Example 03. Two-link endpoint x

Definitions & inputs. l1=l2=0.5 m, θ1=0°, θ2=90°.

  1. Choose the governing model and isolate the requested quantity.

    x=l1cos⁡θ1+l2cos⁡(θ1+θ2)x=l_1\cos\theta_1+l_2\cos(\theta_1+\theta_2)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    x=0.5+0x=0.5+0

  3. Evaluate the expression; the result uses the units shown.

    Result=0.5 m\mathrm{Result}=0.5\ {\rm m}

Interpretation. The endpoint also has y=0.5 m.

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Example 04. Elbow inverse angle

Definitions & inputs. x=y=0.5 m, l1=l2=0.5 m; positive elbow branch.

  1. Choose the governing model and isolate the requested quantity.

    θ2=arccos⁡[(x2+y2−l12−l22)/(2l1l2)]\theta_2=\arccos[(x^2+y^2-l_1^2-l_2^2)/(2l_1l_2)]

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    θ2=arccos⁡0\theta_2=\arccos0

  3. Evaluate the expression; the result uses the units shown.

    Result=1.570796 rad\mathrm{Result}=1.570796\ {\rm rad}

Interpretation. The negative elbow branch is another possible configuration.

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Example 05. Reachability bound

Definitions & inputs. l1=0.4 m, l2=0.3 m.

  1. Choose the governing model and isolate the requested quantity.

    rmax=l1+l2r_{max}=l_1+l_2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    rmax=0.4+0.3r_{max}=0.4+0.3

  3. Evaluate the expression; the result uses the units shown.

    Result=0.7 m\mathrm{Result}=0.7\ {\rm m}

Interpretation. The minimum reachable radius is |0.4−0.3|=0.1 m, before joint limits.

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Example 06. Single-link tangential speed

Definitions & inputs. l=0.5 m, angular rate=2 rad/s.

  1. Choose the governing model and isolate the requested quantity.

    v=lθ˙v=l\dot\theta

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    v=0.5(2)v=0.5(2)

  3. Evaluate the expression; the result uses the units shown.

    Result=1 m s−1\mathrm{Result}=1\ {\rm m\,s}^{-1}

Interpretation. Direction is tangent to the circle, not always along x.

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Example 07. Static joint torque

Definitions & inputs. Perpendicular endpoint force F=10 N, arm l=0.4 m.

  1. Choose the governing model and isolate the requested quantity.

    τ=lF\tau=lF

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    τ=0.4(10)\tau=0.4(10)

  3. Evaluate the expression; the result uses the units shown.

    Result=4 N m\mathrm{Result}=4\ {\rm N\,m}

Interpretation. Nonperpendicular force requires the moment-arm sine factor.

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Example 08. Angular acceleration

Definitions & inputs. Net torque 2 N m, inertia 0.1 kg m².

  1. Choose the governing model and isolate the requested quantity.

    α=τ/J\alpha=\tau/J

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    α=2/0.1\alpha=2/0.1

  3. Evaluate the expression; the result uses the units shown.

    Result=20 rad s−2\mathrm{Result}=20\ {\rm rad\,s}^{-2}

Interpretation. Load torque has already been subtracted.

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Example 09. Motor mechanical power

Definitions & inputs. Torque 0.5 N m, speed 100 rad/s.

  1. Choose the governing model and isolate the requested quantity.

    P=τωP=\tau\omega

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    P=0.5(100)P=0.5(100)

  3. Evaluate the expression; the result uses the units shown.

    Result=50 W\mathrm{Result}=50\ {\rm W}

Interpretation. Electrical input is larger when efficiency is below one.

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Example 10. Gear output torque

Definitions & inputs. Input torque 0.2 N m, speed-reduction ratio 10, efficiency 0.9.

  1. Choose the governing model and isolate the requested quantity.

    τo=ηNτi\tau_o=\eta N\tau_i

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    τo=0.9(10)(0.2)\tau_o=0.9(10)(0.2)

  3. Evaluate the expression; the result uses the units shown.

    Result=1.8 N m\mathrm{Result}=1.8\ {\rm N\,m}

Interpretation. Output speed is one tenth of input under this ratio convention.

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Example 11. Encoder angular step

Definitions & inputs. 4096 resolved counts per revolution.

  1. Choose the governing model and isolate the requested quantity.

    Δθ=2π/N\Delta\theta=2\pi/N

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Δθ=2π/4096\Delta\theta=2\pi/4096

  3. Evaluate the expression; the result uses the units shown.

    Result=0.001533981 rad\mathrm{Result}=0.001533981\ {\rm rad}

Interpretation. Resolved counts already include any quadrature multiplication.

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Example 12. Wheel travel

Definitions & inputs. Radius 0.1 m, rotation 3 revolutions.

  1. Choose the governing model and isolate the requested quantity.

    s=2πrNs=2\pi rN

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    s=2π(0.1)(3)s=2\pi(0.1)(3)

  3. Evaluate the expression; the result uses the units shown.

    Result=1.884956 m\mathrm{Result}=1.884956\ {\rm m}

Interpretation. Slip breaks the rolling-distance relation.

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Example 13. Differential-drive speed

Definitions & inputs. r=0.1 m; ωR=12, ωL=8 rad/s.

  1. Choose the governing model and isolate the requested quantity.

    v=r(ωR+ωL)/2v=r(\omega_R+\omega_L)/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    v=0.1(12+8)/2v=0.1(12+8)/2

  3. Evaluate the expression; the result uses the units shown.

    Result=1 m s−1\mathrm{Result}=1\ {\rm m\,s}^{-1}

Interpretation. This is centre-point forward speed.

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Example 14. Differential-drive yaw rate

Definitions & inputs. Same wheels; track width b=0.5 m.

  1. Choose the governing model and isolate the requested quantity.

    ωz=r(ωR−ωL)/b\omega_z=r(\omega_R-\omega_L)/b

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ωz=0.1(12−8)/0.5\omega_z=0.1(12-8)/0.5

  3. Evaluate the expression; the result uses the units shown.

    Result=0.8 rad s−1\mathrm{Result}=0.8\ {\rm rad\,s}^{-1}

Interpretation. Sign is positive for the defined right-minus-left convention.

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Example 15. Stopping distance

Definitions & inputs. Initial speed 2 m/s; constant braking deceleration magnitude 1 m/s².

  1. Choose the governing model and isolate the requested quantity.

    d=v02/(2a)d=v_0^2/(2a)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    d=22/(2⋅1)d=2^2/(2\cdot1)

  3. Evaluate the expression; the result uses the units shown.

    Result=2 m\mathrm{Result}=2\ {\rm m}

Interpretation. Reaction time adds distance before braking starts.

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Example 16. Triangular move duration

Definitions & inputs. Rest-to-rest distance 1 m, acceleration limit 2 m/s², no velocity cap reached.

  1. Choose the governing model and isolate the requested quantity.

    t=2d/at=2\sqrt{d/a}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    t=21/2t=2\sqrt{1/2}

  3. Evaluate the expression; the result uses the units shown.

    Result=1.414214 s\mathrm{Result}=1.414214\ {\rm s}

Interpretation. Acceleration switches sign halfway through the motion.

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Example 17. Proportional control command

Definitions & inputs. Position error 0.05 rad, Kp=20 N m/rad.

  1. Choose the governing model and isolate the requested quantity.

    u=Kpeu=K_pe

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    u=20(0.05)u=20(0.05)

  3. Evaluate the expression; the result uses the units shown.

    Result=1 N m\mathrm{Result}=1\ {\rm N\,m}

Interpretation. This is a command before actuator limits.

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Example 18. Scalar estimator gain

Definitions & inputs. Prior variance P=4 mm², measurement variance R=1 mm².

  1. Choose the governing model and isolate the requested quantity.

    K=P/(P+R)K=P/(P+R)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    K=4/(4+1)K=4/(4+1)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.8 \mathrm{Result}=0.8\

Interpretation. The sensor receives most of the weight because its variance is smaller.

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Example 19. Posterior position estimate

Definitions & inputs. Prior 10 mm, measurement 12 mm, gain K=0.8.

  1. Choose the governing model and isolate the requested quantity.

    x^+=x^−+K(z−x^−)\hat x^+=\hat x^-+K(z-\hat x^-)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    10+0.8(12−10)10+0.8(12-10)

  3. Evaluate the expression; the result uses the units shown.

    Result=11.6 mm\mathrm{Result}=11.6\ {\rm mm}

Interpretation. The estimate lies between prior and measurement.

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Example 20. Jacobian singularity measure

Definitions & inputs. Planar 2R: l1=l2=0.5 m, θ2=0.

  1. Choose the governing model and isolate the requested quantity.

    det⁡J=l1l2sin⁡θ2\det J=l_1l_2\sin\theta_2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    det⁡J=0.25sin⁡0\det J=0.25\sin0

  3. Evaluate the expression; the result uses the units shown.

    Result=0 m2\mathrm{Result}=0\ {\rm m}^2

Interpretation. A straight elbow is position-singular; a pseudoinverse needs care.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.