Motion, Trajectories, and Robotics | Malone Math

The Central Idea

A motion problem begins by allowing position to depend on time.

In the plane, we may write

\[ \boxed{\mathbf{r}(t)=\langle x(t),y(t)\rangle}. \]

In three dimensions,

\[ \boxed{\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle}. \]

The function \(\mathbf{r}(t)\) is a vector-valued position function. As \(t\) changes, the moving point traces a path through space.

Position

Position identifies where an object is relative to a chosen reference frame.

In two dimensions, a point may have coordinates

\[ (x,y). \]

In three dimensions, it may have coordinates

\[ (x,y,z). \]

Motion occurs when these coordinates change with time.

Displacement

Displacement measures the change in position.

If an object moves from

\[ \mathbf{r}_1 \]

\[ \mathbf{r}_2, \]

then its displacement is

\[ \boxed{\Delta\mathbf{r}=\mathbf{r}_2-\mathbf{r}_1}. \]

Displacement includes both magnitude and direction.

Distance Traveled Is Not Always Displacement

Displacement depends only on the starting and ending positions.

Distance traveled depends on the path taken between them.

If an object moves \(5\) meters east and then \(5\) meters west, its total distance traveled is

\[ 10\text{ m}, \]

but its net displacement is

\[ \boxed{0}. \]

Velocity

Velocity measures the rate of change of position.

\[ \mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle, \]

then the velocity vector is

\[ \boxed{ \mathbf{v}(t) = \mathbf{r}'(t) = \langle x'(t),y'(t),z'(t)\rangle }. \]

Velocity points in the instantaneous direction of motion.

Speed

Speed is the magnitude of velocity.

\[ \boxed{ v= \|\mathbf{v}\| }. \]

In three dimensions,

\[ \boxed{ v= \sqrt{ \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2 } }. \]

Speed has magnitude but no direction.

Acceleration

Acceleration measures the rate of change of velocity.

\[ \boxed{ \mathbf{a}(t) = \mathbf{v}'(t) = \mathbf{r}''(t) }. \]

Acceleration can change the magnitude of velocity, its direction, or both.

Example: Straight-Line Motion

Suppose

\[ \mathbf{r}(t)=\langle 2t,3t\rangle. \]

Then

\[ \mathbf{v}(t)=\langle2,3\rangle. \]

The velocity is constant, so

\[ \mathbf{a}(t)=\langle0,0\rangle. \]

The path is a straight line because the ratio between the coordinates remains constant:

\[ y=\frac{3}{2}x. \]

Parametric Trajectories

A trajectory is naturally described parametrically.

Instead of writing one coordinate directly as a function of another, we write both as functions of time:

\[ \boxed{ x=x(t),\qquad y=y(t) }. \]

This is especially useful when the path loops, reverses, or passes through the same horizontal or vertical coordinate more than once.

Example: Circular Motion

Consider

\[ x(t)=R\cos t, \] \[ y(t)=R\sin t. \]

Then

\[ x^2+y^2 = R^2\cos^2 t + R^2\sin^2 t = R^2. \]

Therefore, the path is the circle

\[ \boxed{x^2+y^2=R^2}. \]

Velocity in Circular Motion

For

\[ \mathbf{r}(t) = \langle R\cos t,R\sin t\rangle, \]

velocity is

\[ \mathbf{v}(t) = \langle -R\sin t,R\cos t\rangle. \]

The position and velocity vectors are perpendicular because

\[ \mathbf{r}(t)\cdot\mathbf{v}(t)=0. \]

Thus, the instantaneous velocity is tangent to the circle.

Trajectory Shape Versus Timing

Two moving objects can follow the same geometric path while moving through it at different rates.

The trajectory

\[ \mathbf{r}_1(t) = \langle\cos t,\sin t\rangle \]

and

\[ \mathbf{r}_2(t) = \langle\cos2t,\sin2t\rangle \]

trace the same unit circle.

The second moves around the circle twice as quickly with respect to the parameter \(t\).

Piecewise Motion

Many practical trajectories are made from several motion segments.

A robot may:

move straight;
turn;
follow an arc;
stop;
reverse or continue along another segment.

A piecewise position function can model this sequence.

Waypoints

A waypoint is a designated position that a path should reach or pass near.

A simple waypoint sequence might be

\[ P_0,\;P_1,\;P_2,\ldots,P_n. \]

A path planner may connect these points with straight segments, smooth curves, or dynamically feasible motion.

Straight-Line Motion Between Waypoints

Suppose a robot moves from

\[ P_0 \]

\[ P_1. \]

A simple interpolation is

\[ \boxed{ \mathbf{r}(t) = (1-t)P_0+tP_1, \qquad 0\le t\le1 }. \]

At \(t=0\), the position is \(P_0\).

At \(t=1\), the position is \(P_1\).

Example: Linear Interpolation

Let

\[ P_0=(2,4), \qquad P_1=(8,10). \]

Then

\[ \mathbf{r}(t) = (1-t)(2,4)+t(8,10). \]

Simplifying,

\[ \boxed{ \mathbf{r}(t) = (2+6t,\;4+6t) }. \]

At \(t=\frac12\), the robot is at the midpoint

\[ \boxed{(5,7)}. \]

Robot Coordinate Frames

A robot may use several coordinate systems at once.

Examples include:

a world frame;
a robot-base frame;
a joint frame;
a tool frame;
a sensor frame;
an object frame.

A point's coordinates depend on which frame is being used.

Local and Global Coordinates

Suppose a point has local coordinates

\[ \mathbf{p}_{\text{local}}. \]

A transformation can convert it into global coordinates:

\[ \boxed{ \mathbf{p}_{\text{global}} = T(\mathbf{p}_{\text{local}}) }. \]

The transformation may include rotation, translation, or both.

Translation

If a local frame is shifted by vector

\[ \mathbf{d} = \langle a,b\rangle, \]

then

\[ \boxed{ \mathbf{p}_{\text{global}} = \mathbf{p}_{\text{local}} + \mathbf{d} }. \]

This changes position without changing orientation.

Rotation

In the plane, a rotation by angle \(\theta\) about the origin can be written with the matrix

\[ \boxed{ R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{bmatrix} }. \]

\[ \mathbf{p} = \begin{bmatrix} x\\ y \end{bmatrix}, \]

then the rotated point is

\[ \boxed{\mathbf{p}'=R(\theta)\mathbf{p}}. \]

Example: Rotating a Point

Rotate

\[ (1,0) \]

\[ 90^\circ. \]

Since

\[ \cos90^\circ=0, \qquad \sin90^\circ=1, \]

we obtain

\[ \boxed{(1,0)\longrightarrow(0,1)}. \]

Rigid Motions in Robotics

A rigid body moves without changing shape.

In planar motion, its pose is determined by:

horizontal position;
vertical position;
orientation angle.

A planar robot pose can therefore be represented as

\[ \boxed{(x,y,\theta)}. \]

Configuration Space

A robot's configuration describes the variables needed to specify its geometric state.

For a planar rigid body,

\[ (x,y,\theta) \]

may be enough.

For a multi-joint robot arm, the configuration may instead be described by joint angles:

\[ \boxed{ \mathbf{q} = \langle \theta_1,\theta_2,\ldots,\theta_n \rangle }. \]

Joint Space and Workspace

Joint space describes a robot using its internal joint variables.

Workspace describes the physical region that the robot or its tool can reach.

A mapping connects the two:

\[ \boxed{ \text{joint configuration} \longrightarrow \text{tool position and orientation} }. \]

A Simple Two-Link Robot Arm

Consider a planar robot arm with link lengths

\[ L_1 \]

and

\[ L_2. \]

Let the first joint angle be \(\theta_1\), and let the second link rotate by additional angle \(\theta_2\) relative to the first.

The end-effector coordinates are

\[ \boxed{ x = L_1\cos\theta_1 + L_2\cos(\theta_1+\theta_2) } \]

and

\[ \boxed{ y = L_1\sin\theta_1 + L_2\sin(\theta_1+\theta_2) }. \]

Example: Two-Link Position

Suppose

\[ L_1=3, \qquad L_2=2, \]

with

\[ \theta_1=0^\circ, \qquad \theta_2=90^\circ. \]

Then

\[ x = 3\cos0^\circ + 2\cos90^\circ = 3, \] \[ y = 3\sin0^\circ + 2\sin90^\circ = 2. \]

Therefore, the end-effector is at

\[ \boxed{(3,2)}. \]

Reachability

A robot can reach only positions allowed by its geometry.

For a two-link planar arm with lengths \(L_1\) and \(L_2\), the farthest possible distance from the base is

\[ \boxed{L_1+L_2}. \]

If the arm can fold fully back on itself, the minimum radial reach is

\[ \boxed{|L_1-L_2|}. \]

Reachable Annulus

Under the simplified two-link model, a target at radial distance \(r\) from the base can be reachable only if

\[ \boxed{ |L_1-L_2| \le r \le L_1+L_2 }. \]

This describes an annular region when all angular orientations are allowed.

Example: Reach Check

A robot arm has link lengths

\[ L_1=4\text{ m}, \qquad L_2=3\text{ m}. \]

Its simplified radial reach satisfies

\[ 1\le r\le7. \]

A target \(6\) meters from the base is potentially reachable.

A target \(8\) meters from the base is not reachable under this model.

Forward Kinematics

Forward kinematics determines the position and orientation of a robot's end effector from known joint variables.

In symbolic form,

\[ \boxed{ \mathbf{x}=f(\mathbf{q}) }. \]

The joint configuration \(\mathbf{q}\) is known, and the spatial pose \(\mathbf{x}\) is calculated.

Inverse Kinematics

Inverse kinematics reverses the problem.

The desired tool position or pose is known, and we seek one or more joint configurations that produce it:

\[ \boxed{ \mathbf{q}=f^{-1}(\mathbf{x}) }. \]

The notation is conceptual: an inverse solution may not be unique, may not exist, or may require solving nonlinear equations.

Multiple Robot Configurations

The same end-effector position may be reachable with different joint configurations.

A planar two-link arm may reach the same point with one elbow orientation above the connecting line and another below it.

Thus, inverse kinematics can have multiple solutions.

Path Versus Trajectory

A path specifies a geometric route through space.

A trajectory adds timing.

The distinction can be summarized as

\[ \boxed{ \text{path}+\text{time law} = \text{trajectory} }. \]

Two robots can follow the same path with different speeds, accelerations, and timing constraints.

Path Length

For a smooth planar trajectory

\[ \mathbf{r}(t) = \langle x(t),y(t)\rangle, \]

the path length from \(t=a\) to \(t=b\) is

\[ \boxed{ s = \int_a^b \sqrt{ \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 } \,dt }. \]

This accumulates the instantaneous distance traveled along the path.

Smooth Motion

A path can be geometrically valid but mechanically undesirable if direction changes abruptly.

Sharp corners can require sudden changes in velocity direction.

Robots and vehicles often use smooth curves so that velocity and acceleration change more gradually.

Tangent Direction

For a smooth trajectory, the velocity vector

\[ \mathbf{v}(t) = \mathbf{r}'(t) \]

points tangent to the path when the velocity is nonzero.

The unit tangent vector is

\[ \boxed{ \mathbf{T} = \frac{\mathbf{v}}{\|\mathbf{v}\|} }. \]

Curvature and Turning

Curvature measures how rapidly the direction of a path changes.

Tight turns have larger curvature than broad turns.

Robotics and vehicle motion often impose limits on curvature because real systems cannot turn arbitrarily sharply.

Minimum Turning Radius

A vehicle with a minimum turning radius \(R_{\min}\) cannot follow a circular arc with radius smaller than that value.

Therefore, a planned turn must satisfy

\[ \boxed{ R\ge R_{\min} }. \]

This is a geometric feasibility constraint.

Obstacles

A motion plan must account for regions the robot cannot occupy.

Obstacles can be represented by geometric sets such as:

circles;
rectangles;
polygons;
polyhedra;
meshes;
more general spatial regions.

Collision avoidance requires the moving system to remain outside forbidden regions.

Point-Robot Collision Check

In a simplified model, a robot may be treated as a point.

If a circular obstacle has center

\[ C=(a,b) \]

and radius

\[ R, \]

then a robot point

\[ P=(x,y) \]

is outside the obstacle when

\[ \boxed{ (x-a)^2+(y-b)^2>R^2 }. \]

Robot Size Matters

Real robots are not points.

A path that keeps the robot center outside an obstacle may still allow the robot body to collide.

One common geometric strategy is to enlarge the obstacle by the robot's clearance radius and then treat the robot center as a point.

Inflated Obstacles

Suppose a circular robot has radius \(r\), and a circular obstacle has radius \(R\).

Collision is avoided when the distance between centers satisfies

\[ \boxed{ d>R+r }. \]

This is equivalent to replacing the obstacle with an inflated radius

\[ \boxed{R_{\text{effective}}=R+r}. \]

Clearance

A path may require more than simple nonintersection.

If a safety clearance \(c\) is required, then the effective obstacle radius becomes

\[ \boxed{ R_{\text{effective}} = R+r+c }. \]

This converts a safety requirement into a geometric constraint.

Example: Circular Obstacle Clearance

A circular obstacle has radius

\[ 2\text{ m}, \]

the robot has radius

\[ 0.5\text{ m}, \]

and the required additional clearance is

\[ 0.25\text{ m}. \]

The effective exclusion radius is

\[ 2+0.5+0.25=2.75\text{ m}. \]

Therefore, the robot center should remain more than

\[ \boxed{2.75\text{ m}} \]

from the obstacle center.

Path Planning

Path planning searches for a route from a starting configuration to a goal configuration while respecting constraints.

A simplified planning problem asks for a path satisfying:

\[ \boxed{ \text{start} \longrightarrow \text{goal} } \]

while avoiding obstacles and remaining inside the robot's feasible motion region.

Shortest Path

If there are no obstacles and motion is unrestricted in a Euclidean plane, the shortest path between two points is the straight segment connecting them.

Its length is

\[ \boxed{ d = \sqrt{ (x_2-x_1)^2 + (y_2-y_1)^2 } }. \]

Obstacles and motion constraints can make the shortest feasible path more complicated.

Planning Around an Obstacle

If a straight-line route intersects an obstacle, the path must change.

A simple planning strategy may introduce intermediate waypoints:

\[ P_{\text{start}} \longrightarrow P_1 \longrightarrow P_2 \longrightarrow P_{\text{goal}}. \]

The total route length is then the sum of the segment lengths.

Dynamic Constraints

A geometrically collision-free path may still be impossible to follow if it requires excessive:

speed;
acceleration;
turning rate;
joint motion;
curvature;
force or torque.

A practical trajectory must satisfy both spatial and motion constraints.

Velocity Limits

If a system has maximum permitted speed

\[ v_{\max}, \]

then a valid trajectory must satisfy

\[ \boxed{ \|\mathbf{v}(t)\| \le v_{\max} } \]

throughout the required motion interval.

Acceleration Limits

If the maximum permitted acceleration magnitude is

\[ a_{\max}, \]

then the trajectory must satisfy

\[ \boxed{ \|\mathbf{a}(t)\| \le a_{\max} }. \]

These limits can protect stability, mechanical components, carried loads, and surrounding systems.

Stopping Distance

In a simplified one-dimensional constant-deceleration model, an object with initial speed \(v\) and braking acceleration magnitude \(a\) has stopping distance

\[ \boxed{ d= \frac{v^2}{2a} }. \]

This relationship shows that stopping distance grows with the square of speed.

Example: Stopping Distance

A robot moves at

\[ 4\text{ m/s} \]

and can decelerate at

\[ 2\text{ m/s}^2. \]

Its simplified stopping distance is

\[ d= \frac{4^2}{2(2)} = \frac{16}{4} = 4. \]

Therefore,

\[ \boxed{d=4\text{ m}}. \]

Relative Motion

When two objects move, their separation depends on their relative position.

\[ \mathbf{r}_A(t) \]

and

\[ \mathbf{r}_B(t) \]

are their positions, then the position of \(B\) relative to \(A\) is

\[ \boxed{ \mathbf{r}_{B/A}(t) = \mathbf{r}_B(t)-\mathbf{r}_A(t) }. \]

Relative Velocity

Differentiating relative position gives relative velocity:

\[ \boxed{ \mathbf{v}_{B/A} = \mathbf{v}_B-\mathbf{v}_A }. \]

This helps analyze approaching objects, moving obstacles, robotic coordination, and collision risk.

Distance Between Moving Objects

If two moving points have positions

\[ \mathbf{r}_A(t) \]

and

\[ \mathbf{r}_B(t), \]

their separation is

\[ \boxed{ d(t) = \| \mathbf{r}_B(t)-\mathbf{r}_A(t) \| }. \]

Collision risk can be studied by determining when \(d(t)\) becomes small enough to violate a required clearance.

Intercept Problems

An intercept occurs when two trajectories reach the same position at the same time.

Mathematically, we seek \(t\) satisfying

\[ \boxed{ \mathbf{r}_A(t)=\mathbf{r}_B(t) }. \]

Matching positions at different times is not enough for a physical interception.

Sampling Motion

Digital systems often observe motion at discrete time steps rather than continuously.

If positions are recorded at

\[ t_0,t_1,t_2,\ldots, \]

then average velocity over one interval is

\[ \boxed{ \mathbf{v}_{\text{avg}} = \frac{ \mathbf{r}(t_{i+1})-\mathbf{r}(t_i) }{ t_{i+1}-t_i } }. \]

Finite-Difference Approximation

If measurements are separated by constant time step \(\Delta t\), velocity can be approximated by

\[ \boxed{ \mathbf{v}(t) \approx \frac{ \mathbf{r}(t+\Delta t)-\mathbf{r}(t) }{ \Delta t } }. \]

This connects continuous motion concepts with numerical and sensor-based robotics.

Sensor-Based Position

Robots may estimate position using:

wheel encoders;
inertial sensors;
cameras;
laser scanners;
satellite positioning;
range measurements;
external tracking systems.

Each measurement contributes information about the robot's spatial state.

Odometry

Odometry estimates position by accumulating measured motion.

In a simplified planar model,

\[ x_{n+1}=x_n+\Delta x_n, \] \[ y_{n+1}=y_n+\Delta y_n. \]

Small measurement errors can accumulate over many steps, causing position drift.

Error Accumulation in Motion

If each motion estimate contains a small error, repeated integration of those estimates can gradually shift the computed position away from the true position.

Robotics therefore often combines motion estimates with independent references to correct accumulated drift.

Feedback

Feedback compares the measured state of a system with the desired state.

A simple position error is

\[ \boxed{ \mathbf{e} = \mathbf{r}_{\text{desired}} - \mathbf{r}_{\text{actual}} }. \]

Control actions can then attempt to reduce this error.

Tracking a Trajectory

If the desired trajectory is

\[ \mathbf{r}_d(t) \]

and the actual trajectory is

\[ \mathbf{r}_a(t), \]

then the tracking error is

\[ \boxed{ \mathbf{e}(t) = \mathbf{r}_d(t)-\mathbf{r}_a(t) }. \]

Good tracking keeps the error appropriately small for the task.

Common Error: Confusing Position and Displacement

Position describes location relative to a reference frame.

Displacement describes the change between two positions.

\[ P_1=(2,3) \]

and

\[ P_2=(7,5), \]

then the displacement is

\[ \boxed{(5,2)}, \]

not \((7,5)\).

Common Error: Confusing Speed and Velocity

Speed is a scalar magnitude.

Velocity is a vector.

Two objects can have equal speed while moving in different directions and therefore have different velocities.

Common Error: Ignoring the Time Parameter

A geometric path alone does not tell us where an object is at a particular time.

Trajectory analysis requires both the path and its parameterization in time.

Common Error: Treating the Robot as a Point

Point models are useful simplifications, but real robots occupy area or volume.

Collision checking must account for robot geometry or use appropriately enlarged obstacle regions.

Common Error: Planning Only for Position

A robot may reach the correct point with the wrong orientation.

For many tasks, the required state includes both position and orientation:

\[ \boxed{(x,y,\theta)} \]

in a planar rigid-body model.

Common Error: Ignoring Motion Limits

A shortest geometric path may require turns, speeds, or accelerations that the physical system cannot execute.

Feasible motion must respect both geometry and dynamics.

Common Error: Accumulating Small Position Errors Without Correction

Repeated motion estimates can drift because small errors accumulate.

Independent references, feedback, and repeated localization checks help keep the estimated position aligned with the physical system.

A Reliable Procedure for Trajectory Problems

Identify the coordinate frame.
Write the position as a function of time.
Differentiate to obtain velocity when needed.
Differentiate again to obtain acceleration when needed.
Check units.
Determine whether the question concerns path, timing, speed, direction, or all of them.
Apply geometric constraints such as obstacles or reach limits.
Check physical constraints such as maximum speed or turning radius.
Verify that the result is spatially and physically reasonable.

A Reliable Procedure for Basic Robot Reach Problems

Identify the robot's link geometry.
Identify the desired target position.
Measure or calculate the target's distance from the base.
Compare that distance with the robot's reach limits.
Use forward or inverse kinematics as appropriate.
Check joint limits and orientation requirements.
Check obstacle clearance along the motion path.

Example: Full Robot Motion Workflow

A circular mobile robot of radius

\[ 0.40\text{ m} \]

must move from

\[ A=(0,0) \]

\[ B=(6,8). \]

First calculate the straight-line path length:

\[ d= \sqrt{6^2+8^2} = 10\text{ m}. \]

Suppose a circular obstacle of radius

\[ 1.0\text{ m} \]

lies near the route, and the required safety clearance is

\[ 0.20\text{ m}. \]

The obstacle should be treated as having effective radius

\[ R_{\text{effective}} = 1.0+0.40+0.20 = 1.60\text{ m}. \]

The straight path is acceptable only if the robot center remains more than \(1.60\) meters from the obstacle center throughout the motion.

If not, the path must be rerouted around the expanded obstacle region.

Practice 1: Displacement

A robot moves from

\[ (3,4) \]

\[ (9,1). \]

Find its displacement vector.

Practice 2: Velocity

Given

\[ \mathbf{r}(t) = \langle 3t^2,4t\rangle, \]

find the velocity vector.

Practice 3: Acceleration

Using the position function from Practice 2, find the acceleration vector.

Practice 4: Speed

\[ \mathbf{v} = \langle 6,8\rangle, \]

find the speed.

Practice 5: Linear Interpolation

A robot moves linearly from

\[ P_0=(0,2) \]

\[ P_1=(10,6). \]

Find its position at \(t=0.25\) using

\[ \mathbf{r}(t)=(1-t)P_0+tP_1. \]

Practice 6: Reachability

A planar two-link arm has

\[ L_1=5, \qquad L_2=2. \]

Give the simplified minimum and maximum radial reach.

Practice 7: Effective Obstacle Radius

A circular obstacle has radius

\[ 1.5\text{ m}, \]

a circular robot has radius

\[ 0.4\text{ m}, \]

and required clearance is

\[ 0.3\text{ m}. \]

Find the effective exclusion radius.

Practice 8: Stopping Distance

A robot moves at

\[ 6\text{ m/s} \]

and can decelerate at

\[ 3\text{ m/s}^2. \]

Find the simplified stopping distance.

Practice 9: Relative Position

Robot \(A\) is at

\[ (2,5), \]

and robot \(B\) is at

\[ (8,9). \]

Find the position of \(B\) relative to \(A\).

Practice 10: Identify the Error

A student plans a straight-line robot path that does not pass through an obstacle, but the robot body clips the obstacle.

Explain the modeling error.

Practice Answers

Practice 1

\[ \Delta\mathbf{r} = (9-3,\;1-4). \]

Therefore,

\[ \boxed{\Delta\mathbf{r}=(6,-3)}. \]

Practice 2

\[ \mathbf{v}(t) = \mathbf{r}'(t) = \langle6t,4\rangle. \]

Therefore,

\[ \boxed{\mathbf{v}(t)=\langle6t,4\rangle}. \]

Practice 3

\[ \mathbf{a}(t) = \mathbf{v}'(t) = \langle6,0\rangle. \]

Therefore,

\[ \boxed{\mathbf{a}(t)=\langle6,0\rangle}. \]

Practice 4

\[ v= \sqrt{6^2+8^2} = \sqrt{100}. \]

Therefore,

\[ \boxed{v=10}. \]

Practice 5

\[ \mathbf{r}(0.25) = 0.75(0,2)+0.25(10,6). \] \[ \mathbf{r}(0.25) = (0,1.5)+(2.5,1.5). \]

Therefore,

\[ \boxed{(2.5,3)}. \]

Practice 6

Minimum simplified radial reach:

\[ |5-2|=3. \]

Maximum radial reach:

\[ 5+2=7. \]

Therefore,

\[ \boxed{3\le r\le7}. \]

Practice 7

\[ R_{\text{effective}} = 1.5+0.4+0.3 = 2.2. \]

Therefore,

\[ \boxed{2.2\text{ m}}. \]

Practice 8

\[ d= \frac{6^2}{2(3)} = \frac{36}{6}. \]

Therefore,

\[ \boxed{d=6\text{ m}}. \]

Practice 9

\[ \mathbf{r}_{B/A} = (8-2,\;9-5). \]

Therefore,

\[ \boxed{(6,4)}. \]

Practice 10

The student treated the robot as though only its center point mattered.

A real robot occupies area or volume, so obstacle clearance must account for the robot's physical size.

One simplified solution is to enlarge the obstacle by the robot's effective radius and plan the center-point path around the enlarged region.

What to Remember

Position becomes a function of time in motion problems.
Displacement is the change in position, while distance traveled depends on the path.
Velocity is the rate of change of position, and speed is the magnitude of velocity.
Acceleration is the rate of change of velocity.
Parametric equations describe trajectories naturally.
A path describes geometry; a trajectory adds timing.
Robots use multiple coordinate frames connected by transformations.
Forward kinematics maps joint variables to spatial pose, while inverse kinematics seeks joint variables for a desired pose.
Reachability depends on the geometry and limits of the mechanism.
Collision avoidance must account for the robot's physical size, not merely its center point.
Path planning must satisfy obstacle, reach, speed, acceleration, and turning constraints.
Relative motion is analyzed by subtracting positions and velocities.
Sensor-based motion estimates can accumulate error and benefit from feedback and independent position references.

The central relationship is:

\[ \boxed{ \text{geometry} + \text{time} + \text{constraints} \longrightarrow \text{feasible motion} }. \]

Where We Go Next

Robotics uses geometry to move physical systems through space. Computer graphics and animation use many of the same mathematical ideas to move, transform, and display virtual objects.

The next lesson, Computer Graphics and Animation, develops spatial transformations, coordinate systems, projection, interpolation, camera geometry, and motion in visual computing.