Calculus

Differentiation in Kinematics

Robotic motion is often defined by a position function q(t)q(t) of joint angles or Cartesian coordinates over time. Differentiation yields:

  • Velocity

qË™(t)=ddtq(t)\dot{q}(t) = \frac{d}{dt}q(t)
  • Acceleration

Example: for a single-joint rotary robot moving from angle θ0θ0 to θfθf via a cubic trajectory, the velocity and acceleration profiles are polynomials in tt.

theta(t)=a0+a1t+a2t2+a3t3theta(t) = a_0 + a_1 t + a_2 t^2 + a_3 t^3
theta(t)=a1+2a2 t+3a3 t2theta(t) = a_1 + 2a_2\,t + 3a_3\,t^2
theta(t)=2a2+6a3 ttheta(t) = 2a_2 + 6a_3\,t

Integration in Motion and Sensing

Integration accumulates rates into displacements or sensor estimates:

  • Position from velocity: q(t)=q(t0)+∫t0tqË™(Ï„) dÏ„q(t)=q(t0)+∫t0tqË™(Ï„)dÏ„

  • State estimation (e.g., odometry): integrating wheel velocities to track robot pose.

Example: integrating a constant acceleration aa gives velocity and displacement:

v(t)=v0+a tv(t) = v_0 + a\,t
s(t)=s0+v0 t+12 a t2s(t) = s_0 + v_0\,t + \tfrac12\,a\,t^2

Trajectory Planning

Selecting smooth paths between waypoints requires piecewise-polynomial fits that ensure continuous position, velocity, and sometimes acceleration. A common choice is the cubic segment between times titi and ti+1ti+1:

qi(τ)=C0+C1τ+C2τ2+C3τ3,τ∈[0, T]q_i(\tau) = C_0 + C_1 \tau + C_2 \tau^2 + C_3 \tau^3,\quad \tau\in[0,\,T]

With boundary conditions

qi(0)=q(ti)qi(0)=q(ti)
q˙i(0)=q˙(ti)q˙i(0)=q˙(ti)
qi(T)=q(ti+1)qi(T)=q(ti+1)
q˙i(T)=q˙(ti+1)q˙i(T)=q˙(ti+1)
qi(0)=q(ti)qi(0)=q(ti)
q˙i(0)=q˙(ti)q˙i(0)=q˙(ti)
qi(T)=q(ti+1)qi(T)=q(ti+1)
q˙i(T)=q˙(ti+1)q˙i(T)=q˙(ti+1)

Dynamics via Lagrange’s Equations

Robot dynamics relate joint torques ττ, positions qq, and accelerations q¨q¨. The Euler–Lagrange formulation yields:

M(q) q¨+C(q,q˙) q˙+g(q)=τM(q)\,\ddot q + C(q,\dot q)\,\dot q + g(q) = \tau

where

  • M(q)M(q) is the mass (inertia) matrix,

  • C(q,qË™)C(q,qË™) contains Coriolis and centrifugal terms,

  • g(q)g(q) is the gravity vector.

Control System Design

Calculus underlies feedback controllers and optimizations:

  • PID control uses proportional, integral, and derivative terms on tracking error e(t)e(t):

u(t)=KPe(t)+KI∫0te(τ) dτ+KDddte(t)u(t) = K_P e(t) + K_I \int_0^t e(\tau)\,d\tau + K_D \tfrac{d}{dt}e(t)
  • Linear Quadratic Regulator (LQR) solves continuous-time algebraic Riccati equations to minimize a quadratic cost.

Perception and Computer Vision

In vision, image coordinates x(u,v)x(u,v) vary with time as cameras or objects move. Optical flow computes image velocity:

I˙+∇I⋅v=0\dot I + \nabla I \cdot \mathbf{v} = 0

where I˙I˙ is the temporal image derivative, ∇I∇I its spatial gradient, and vv the pixel velocity.

Calculus thus weaves through every phase of robotics: from physical motion and sensor fusion to high-level planning and control, allowing robots to move smoothly, react reliably, and perceive accurately.

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