11.4 - Kinematic Analysis

Animations

Fusion360 motion study illustrating how servo inputs drive the jumping mechanism through a defined trajectory in input space. Revolute 4 represents the top servo input, and revolute 1 represents the bottom servo’s rotational input through the entire jumping motion profile. The vertical sliding joint is included to simulate the desired height of the vertically-constrained servo body.

Fusion360 motion study showing how smooth variation of servo inputs produces a controlled foot trajectory. Revolute 4 represents the top servo input, and revolute 1 represents the bottom servo’s rotational input through the entire motion profile. The vertical sliding joint is not required in this motion study, as the vertically constrained servo body experiences negligible vertical displacement.

Kinematic Diagram and Link Lengths

Kinematic diagram of our mechanism showing two coupled four-bar sub chains (O2 A B O1 and O1 C D E) driven by independent servo motors and coupled by a ternary link.

Corresponding lengths of each link in our mechanism. The system is best described as a two-servo planar six-bar leg mechanism with a four-bar input stage and a coupled four-bar lower-leg stage.

Linkage Orientation Throughout Jumping Motion

1. Mechanism at zero configuration (rest), where both servos are at 0°. The vertical slider displacement is also set to zero.

2. Mechanism at full crouch position, where servo 1 is at 22.2° and servo 2 is at -30° orientation. The vertical slider displacement is -82.5mm from zero configuration.

3. Mechanism at takeoff position just before foot leaves the ground, where servo 1 is at 0° and servo 2 is at 45°. The vertical slider displacement is 34.60mm from zero configuration.

4. Mechanism at maximum height of sliding body, where servo 1 is at -15° and servo 2 is at 45° orientation. The vertical slider displacement is 64.60mm from zero configuration.

5. Mechanism orientation around the halfway point back down from maximum height, where servo 1 is at 10° and servo 2 is at -5°. The vertical slider displacement is 49.60mm from zero configuration.

6. Mechanism at landing position, where servo 1 is at 15° and servo 2 is at -15°. The vertical slider displacement is -43.25mm from zero configuration.

Consolidated table of each servo shaft and individual link’s orientation at the various phases throughout the jumping motion. These are the values used for further kinematic analysis involving velocity, acceleration, and force.

Consolidated table representing the sliding body’s vertical displacement at each phase relative to its position at zero configuration.

Position Analysis

Kinematic analysis was performed using two coordinate frames. The local body frame is fixed at O1- (the primary servo/hip pivot) to describe the relative motion of the links. Joint positions were calculated from link lengths and link orientations in the local frame first. A separate global ground frame was then used to capture the absolute jumping motion. By adding the vertical slider displacement s(t) to the calculated local positions, coordinates of the local body frame are translated into the global ground frame.

For local coordinates fixed at O1 = (0,0):

For global coordinates with ground reference:

All joint positions can be geometrically determined in this manner; however, only joints E and F are required to describe our system’s output motion. The remaining joints serve as internal constraints, and do not directly affect the performance metrics being evaluated.

Plugging our simulated values for link orientation and slider vertical displacement into the equations above for each phase in the jumping motion:

MATLAB-Generated Position Plots

3D plot of the foot’s vertical position as a function of two independent servo inputs. The traectory shows how combined actuation of θ1 and θ2 produces the jumping motion, with minimum height at full crouch and maximum height during the airborne phase. The nonlinear path reflects the coupled, two-input nature of the mechanism.

This 3D plot illustrates how the prismatic joint displacement relative to its rest position varies with θ1 and θ2, converting internal linkage motion into vertical body translation. Maximum upward displacement occurs near the airborne phase, while the lowest position corresponds to full crouch during energy storage.

Servo input angles as functions of time defining the prescribed motion profile for position analysis. These input trajectories for θ1 and θ2, extracted from the motion study, serve as the driving parameters for the kinematic model, determining the corresponding link orientations and resulting positions of key points such as the foot and slider throughout the jumping cycle.

Velocity Analysis

Moving beyond position analysis, each joint’s velocity was computed from position data using a central difference approximation, which approximates the derivative based on neighboring position values. Rather than assuming uniform time steps to use the simplified equation, the timing of each phase was obtained directly from the Autodesk Fusion motion study, where motion is defined as a perentage of the total cycle. When converted to actual time values, these percentages result in non-uniform time intervals between phases. The central difference method was then applied using these variable time steps, allowing the velocity calculations to accurately reflect our mechanism’s dynamics.

Central difference method for uniform time distribution

Central difference method for non-uniform time intervals

MATLAB-Generated Velocity Plots:

This plot illustrates how combined variations in θ1 and θ2 produce nonlinear changes in the foot’s vertical velocity, with peak upward velocity occurring near take-off and peak downward velocity during landing, highlighting the dynamic response of the end effector.

The prismatic joint velocity reflects how rotational inputs are converted into vertical body motion, with rapid upward acceleration during take-off and significant downward velocity during landing, indicating efficient energy transfer through the mechanism.

These profiles show the rate of change of θ1 and θ2, revealing phase-dependent actuation where rapid changes in angular velocity correspond to key events such as take-off and landing, driving the overall kinematic behavior of the system.

Acceleration Analysis

The acceleration of each point was calculated from the velocity data using a numerical differentiation approach consistent with the non-uniform time stepping extracted from the Fusion 360 motion study. Similar to our velocity analysis, the acceleration of each joint was obtained by applying a central difference formula to the velocity data using the actual time vector, rather than assuming uniform time intervals. This ensures that variations in phase timing are properly captured in the second derivative of position; therefore, the resulting acceleration values reflect the true dynamic behavior of the mechanism, including rapid changes during key events such as take-off and landing. By basing the acceleration calculations directly on the previously computed velocity profiles, consistency is maintained between position, velocity, and acceleration, providing an accurate representation of how the mechanism responds to the prescribed servo inputs.

Acceleration central difference method for uniform time intervals.

Acceleration central difference method for non-uniform time intervals.

MATLAB-Generated Acceleration Plots

This plot highlights the highly nonlinear relationship between actuator inputs and end dynamics, with large positive acceleration during take-off and significant negative acceleration during landing, corresponding to rapid changes in motion direction and impact forces.

The prismatic joint experiences large accelerations during key phases of the motion, particularly during energy storage (crouch) and release (take-off), demonstrating efficient transmission of actuator input into vertical body dynamics.

These profiles show rapid changes in actuator acceleration across motion phases, with sharp peaks corresponding to transitions such as take-off and landing, indicating the dynamic input required to drive the mechanism through its prescribed trajectory.

Force Analysis

The force analysis is directly derived from the previously computed acceleration of the system using Newton’s Second Law. Specifically, the vertical ground reaction force is determined from the acceleration of the body (slider), which represents the mass being accelerated during the jumping motion. By converting the body acceleration into physical units and combining it with gravitational effects, the net ground reaction force can be expressed as F=m(g+a). This establishes a direct relationship between the kinematic analysis and the dynamic response of the system, where rapid changes in acceleration (particularly during take-off and landing) result in corresponding variations in ground reaction force.

This plot illustrates how variations in θ1 and θ2 influence the force transmitted between the mechanism and the ground, with increased force during take-off corresponding to upward body acceleration and reduced force near maximum height where acceleration approaches zero.

MATLAB CODE:

clear; clc; close all

phase = ["Rest","Full Crouch","Take-off","Max Height", ...

"Halfway Down","Landing","Back to Rest"];

n = length(phase);

T_total = 2.2; % total motion time, seconds

% Non-uniform phase timing from Fusion 360 motion study

% These percentages define when each phase occurs in the total jump cycle

motionPercent = [0, 40, 55, 60, 75, 85, 100];

% Convert Fusion motion percentage into actual time values

t = T_total * motionPercent/100;

%% =========================

% LINK LENGTHS, mm

%% =========================

L_O2A = 24;

L_AB = 28.284;

L_O1B = 24;

L_O1C = 30;

L_O1E = 110;

L_CD = 110;

L_ED = 30;

L_EF = 110;

%% =========================

% SERVO INPUT ANGLES, degrees

%% =========================

theta1_deg = [0, 22.2, 0, -15, 10, 15, 0];

theta2_deg = [0, -30, 45, 45, -5, -15, 0];

theta1 = deg2rad(theta1_deg);

theta2 = deg2rad(theta2_deg);

%% =========================

% LINK ORIENTATIONS, degrees

%% =========================

alpha_O1E_deg = [-135, -157.2, -135, -120, -145, -150, -135];

alpha_EF_deg = [-45, -15, -90, -90, -50, -30, -45];

alpha_O1E = deg2rad(alpha_O1E_deg);

alpha_EF = deg2rad(alpha_EF_deg);

%% =========================

% SLIDER DISPLACEMENT, mm

%% =========================

s = [0, -82.5, 34.6, 64.6, 49.6, -43.25, 0];

%% =========================

% POSITION ANALYSIS

%% =========================

xE_rel = L_O1E*cos(alpha_O1E);

zE_rel = L_O1E*sin(alpha_O1E);

xF_rel = xE_rel + L_EF*cos(alpha_EF);

zF_rel = zE_rel + L_EF*sin(alpha_EF);

xE_abs = xE_rel;

zE_abs = zE_rel + s;

xF_abs = xF_rel;

zF_abs = zF_rel + s;

%% =========================

% VELOCITY ANALYSIS

% Non-uniform time stepping using Fusion time vector

%% =========================

vF_x = gradient(xF_abs, t);

vF_z = gradient(zF_abs, t);

vF_mag = sqrt(vF_x.^2 + vF_z.^2);

v_body = gradient(s, t);

omega1 = gradient(theta1, t); % rad/s

omega2 = gradient(theta2, t); % rad/s

%% =========================

% ACCELERATION ANALYSIS

% Non-uniform time stepping using Fusion time vector

%% =========================

aF_x = gradient(vF_x, t);

aF_z = gradient(vF_z, t);

aF_mag = sqrt(aF_x.^2 + aF_z.^2);

a_body = gradient(v_body, t);

alpha1 = gradient(omega1, t); % rad/s^2

alpha2 = gradient(omega2, t); % rad/s^2

%% =========================

% FORCE ANALYSIS

%% =========================

m = 0.250; % kg, change if needed

g = 9.81; % m/s^2

a_body_m = a_body/1000; % convert mm/s^2 to m/s^2

F_ground = m*(g + a_body_m); % N

%% =========================

% RESULTS TABLE

%% =========================

results = table(phase', motionPercent', t', theta1_deg', theta2_deg', s', ...

xF_abs', zF_abs', vF_z', aF_z', v_body', a_body', F_ground', ...

'VariableNames', {'Phase','MotionPercent','Time_s','Theta1_deg','Theta2_deg', ...

'Slider_mm','xF_abs_mm','zF_abs_mm','vFz_mm_s', ...

'aFz_mm_s2','vBody_mm_s','aBody_mm_s2','Fground_N'});

disp(results)

%% =========================

% 3D INPUT-OUTPUT PLOTS

%% =========================

plotInputOutput3D(theta1_deg, theta2_deg, zF_abs, phase, ...

'z_F Absolute (mm)', 'Foot Vertical Position vs Servo Inputs');

plotInputOutput3D(theta1_deg, theta2_deg, s, phase, ...

'Slider Position s(t) (mm)', 'Slider Position vs Servo Inputs');

plotInputOutput3D(theta1_deg, theta2_deg, vF_z, phase, ...

'v_{F,z} (mm/s)', 'Foot Vertical Velocity vs Servo Inputs');

plotInputOutput3D(theta1_deg, theta2_deg, v_body, phase, ...

'v_{body} (mm/s)', 'Body/Slider Velocity vs Servo Inputs');

plotInputOutput3D(theta1_deg, theta2_deg, aF_z, phase, ...

'a_{F,z} (mm/s^2)', 'Foot Vertical Acceleration vs Servo Inputs');

plotInputOutput3D(theta1_deg, theta2_deg, a_body, phase, ...

'a_{body} (mm/s^2)', 'Body/Slider Acceleration vs Servo Inputs');

plotInputOutput3D(theta1_deg, theta2_deg, F_ground, phase, ...

'Ground Reaction Force (N)', 'Ground Reaction Force vs Servo Inputs');

%% =========================

% SERVO INPUT KINEMATICS VS TIME

%% =========================

figure

plot(t, theta1_deg, 'o-', 'LineWidth', 2)

hold on

plot(t, theta2_deg, 'o-', 'LineWidth', 2)

grid on

xlabel('Time (s)')

ylabel('Servo Angle (deg)')

legend('\theta_1','\theta_2')

title('Servo Input Angles vs Time')

figure

plot(t, omega1, 'o-', 'LineWidth', 2)

hold on

plot(t, omega2, 'o-', 'LineWidth', 2)

grid on

xlabel('Time (s)')

ylabel('Angular Velocity (rad/s)')

legend('\omega_1','\omega_2')

title('Servo Angular Velocity vs Time')

figure

plot(t, alpha1, 'o-', 'LineWidth', 2)

hold on

plot(t, alpha2, 'o-', 'LineWidth', 2)

grid on

xlabel('Time (s)')

ylabel('Angular Acceleration (rad/s^2)')

legend('\alpha_1','\alpha_2')

title('Servo Angular Acceleration vs Time')

%% =========================

% LOCAL FUNCTION

%% =========================

function plotInputOutput3D(theta1_deg, theta2_deg, zData, phase, zLabelText, plotTitle)

n = length(theta1_deg);

arrowLen = 8; % adjust between 4 and 12 if arrows look too big/small

figure

plot3(theta1_deg, theta2_deg, zData, 'o-', 'LineWidth', 2)

grid on

hold on

for i = 1:n-1

dx = theta1_deg(i+1) - theta1_deg(i);

dy = theta2_deg(i+1) - theta2_deg(i);

dz = zData(i+1) - zData(i);

mag = sqrt(dx^2 + dy^2 + dz^2);

if mag > 0

quiver3(theta1_deg(i), theta2_deg(i), zData(i), ...

arrowLen*dx/mag, arrowLen*dy/mag, arrowLen*dz/mag, ...

0, 'r', 'LineWidth', 1.5, 'MaxHeadSize', 1.5)

end

for i = 1:n

text(theta1_deg(i), theta2_deg(i), zData(i), ...

[' ', char(phase(i))], 'FontSize', 8)

end

xlabel('\theta_1 (deg)')

ylabel('\theta_2 (deg)')

zlabel(zLabelText)

title(plotTitle)

axis tight

view(45,30)

end