Lesson 2: Position Analysis of Planar Linkages

Mobility analysis confirmed that the four-bar linkage, slider-crank, scissor lift, and toggle clamp each move with a single degree of freedom. That tells you one input controls them, but not where that input puts the output. Turn the crank of a four-bar by a known angle and the coupler and follower take up specific angles that the geometry alone decides. Position analysis finds those angles. It is the problem you must solve before any velocity, acceleration, or force analysis, because all of those are built on top of the positions. In this lesson you write the vector loop equation for each mechanism and solve it in closed form, then check the answer in the simulator. #PositionAnalysis #VectorLoops #Freudenstein

Learning Objectives

By the end of this lesson, you will be able to:

1. Formulate the vector loop equation for any closed planar linkage
2. Solve the four-bar position problem in closed form with the Freudenstein equation
3. Classify mechanisms with the Grashof condition and handle open and crossed assembly modes
4. Locate coupler-point paths and limit (dead-center) positions, verifying each against a simulator

Real-World System Problem: Predicting Where the Output Goes

A windshield wiper must sweep a precise arc and stop short of the A-pillar. An engine piston must reach top-dead-center exactly when the crank does. A scissor-lift platform must reach a target height and stay level. In every case the designer commits to link lengths and then has to predict, before any metal is cut, exactly where the output sits for every input position. Guessing is not an option, because a few millimetres of error can mean a wiper that hits the trim or a piston that collides with a valve.

The Position Problem

Position analysis answers one question for a single-degree-of-freedom mechanism:

Engineering Question: Given the input link angle, what are the angles and coordinates of every other link?

For an open serial chain (a robot arm), this is direct: add the links nose to tail. For a closed chain (the four mechanisms of this course), the links form a loop that must close on itself, and that closure condition is what we solve.

Why Position Analysis Comes First

Fundamental Theory: The Vector Loop Method

Links as Vectors That Close a Loop

Vector Loop Principle

Treat each link as a vector pointing from one joint to the next. In any closed kinematic loop, walking around the links and back to the start gives zero net displacement:

This single geometric fact turns a drawing into algebra. Each vector contributes a cosine term to the x-equation and a sine term to the y-equation, giving two scalar equations per loop.

The power of the method is that it is the same for every linkage. You label the links as vectors, write the loop sum, split it into x and y components, mark which angles are known and which are unknown, and solve. The four-bar gives two equations in two unknown angles. The slider-crank gives two equations in one unknown angle and one unknown distance. The procedure never changes; only the bookkeeping does.

The Four-Bar Loop in Component Form

We use the same link labels as the simulator: link is the input crank, the coupler, the follower, and the ground. Place the origin at the crank pivot, the ground link along the positive x-axis, and measure all angles counterclockwise from it.

Four-Bar Vector Loop

X-component:

Y-component:

Known: link lengths and the input angle . Unknown: the coupler angle and the follower angle .

These are two nonlinear equations in two unknowns. The next section solves them in closed form, so no guessing or iteration is needed.

Application 1: Four-Bar Position by the Freudenstein Equation

This is the central worked example of the lesson. We solve the four-bar position problem exactly, then read the same angles off the simulator.

Simulator and hands-on lab

Open the Four-Bar Linkage Simulator

Hands-on lab: Continue in the Four-Bar Linkage Experiments lab. Experiment 5 (coupler-curve exploration) extends the position analysis below to the path traced by a coupler point.

Step 1: Draw the Space Diagram and Measure

Before solving algebraically, find the position graphically: draw the linkage to scale and measure the angles with a protractor.

Click to reveal the graphical position solution

1. Ground and crank. Draw mm horizontally. From draw the crank at , length mm. ✅

2. Intersect the arcs. With centre and radius mm, and centre and radius mm, strike two arcs. The upper intersection (open assembly) is the coupler-follower joint . ✅

3. Measure. With a protractor and . The Freudenstein solution below gives the exact and , confirming the drawing. ✅

Step 2: Eliminate the Coupler Angle

Click to reveal the elimination of

1. Isolate the coupler terms in both component equations:

   ✅

2. Square both equations and add. Because , the coupler angle disappears, leaving one equation in :

   ✅

Step 3: Form the Freudenstein Equation

Click to reveal the Freudenstein constants

1. Divide through by and group the constant terms. The result is the Freudenstein equation:

   ✅

2. Define the three constants from the link lengths only:

   ✅

3. Evaluate for , , , :

   ✅

Step 4: Solve for the Follower Angle

Click to reveal the closed-form solution for

1. Expand and substitute the half-angle . This converts the equation into a quadratic with:

   ✅

2. Evaluate at (, ):

   ✅

3. Solve the quadratic. The two roots give the two assembly modes:

   ✅

4. Open mode (minus sign):

   ✅

   The crossed mode (plus sign) gives the other branch, discussed in Step 5.

Step 5: Solve for the Coupler Angle

Click to reveal the closed-form solution for

1. Eliminate instead (square-and-add the follower terms) to get a second Freudenstein form in , with:

   ✅

2. Quadratic coefficients:

   ✅

3. Solve (open mode, minus sign):

   ✅

4. Check the result in the original loop equation (, , ):

   ✅

Step 6: Sweep the Crank and Verify in the Simulator

Click to reveal the full-rotation table and verification

1. Repeat the solution for crank angles across a full turn (open mode). A short Python script automates it: ✅

   import numpy as np
   
   def four_bar_position(a, b, c, d, theta2_deg, mode="open"):
       """Closed-form four-bar position. a=crank, b=coupler, c=follower, d=ground."""
       t2 = np.radians(theta2_deg)
       K1, K2 = d/a, d/c
       K3 = (a**2 - b**2 + c**2 + d**2) / (2*a*c)
       A = np.cos(t2) - K1 - K2*np.cos(t2) + K3
       B = -2*np.sin(t2)
       C = K1 - (K2 + 1)*np.cos(t2) + K3
       s = -1 if mode == "open" else 1
       t4 = 2*np.arctan2(-B + s*np.sqrt(B**2 - 4*A*C), 2*A)
   
       K4 = d/b
       K5 = (c**2 - d**2 - a**2 - b**2) / (2*a*b)
       D = np.cos(t2) - K1 + K4*np.cos(t2) + K5
       E = -2*np.sin(t2)
       F = K1 + (K4 - 1)*np.cos(t2) + K5
       t3 = 2*np.arctan2(-E + s*np.sqrt(E**2 - 4*D*F), 2*D)
       return np.degrees(t3) % 360, np.degrees(t4) % 360
   
   for th2 in range(0, 121, 30):
       th3, th4 = four_bar_position(40, 120, 80, 100, th2)
       print(f"theta2={th2:3d}  theta3={th3:6.1f}  theta4={th4:6.1f}")

2. Results (open mode): ✅

   Crank	Coupler	Follower
   0°	36.3°	62.7°
   30°	22.4°	55.3°
   60°	18.4°	64.9°
   90°	18.9°	80.3°
   120°	22.0°	96.3°

3. Verify in the simulator. Set , , , , choose the Open assembly, and drive the crank to each angle. The simulator’s reported and match the table. ✅

Theory: Grashof Classification

Before solving a four-bar, you can predict how it will move from the link lengths alone. The Grashof condition compares the shortest and longest links to the other two.

The Grashof Condition

Let be the shortest link, the longest, and the remaining two. The linkage is Grashof (at least one link can fully rotate) when:

For our default four-bar: (crank), (coupler), and . Since , it is Grashof, and because the shortest link is the crank, it is a crank-rocker: the crank fully rotates while the follower rocks.

• Grashof (s + l < p + q)
• Non-Grashof (s + l > p + q)
• Change-point (s + l = p + q)

At least one link makes a full revolution. Which one depends on which link is the ground:

• Shortest link is the crank or follower: crank-rocker (input rotates, output rocks)
• Shortest link is the ground: double-crank (both side links rotate fully)
• Shortest link is the coupler: double-rocker (both side links rock, coupler rotates)

Application 2: Slider-Crank Position

The slider-crank loop has one unknown angle (the connecting rod) and one unknown distance (the slider position), so its position solution is a single closed-form expression rather than a quadratic.

Simulator and hands-on lab

Open the Crank-Slider Simulator

Hands-on lab: Continue in the Crank-Slider Experiments lab. Experiment 1 (baseline kinematic profile) plots the slider displacement you derive below across a full crank rotation.

Step 1: Draw the Space Diagram and Measure

Click to reveal the graphical position solution

1. Centre-line and crank. Draw the slider centre-line through . Draw the crank at , length mm. ✅

2. Swing the rod. With centre and radius mm, cut the centre-line at , the piston. ✅

3. Measure. The piston is mm from and the rod angle , matching the closed-form values below. ✅

Step 2: Write and Solve the Loop

Click to reveal the slider-crank position solution

1. Loop in components. With the slider on the x-axis at distance from the crank pivot, crank angle , rod angle below the axis, and vertical offset :

   ✅

2. Solve the y-equation for the rod angle :

   ✅

3. Substitute into the x-equation for the slider position (eliminating with ):

   ✅

Step 3: Evaluate and Verify

Click to reveal the numbers and simulator check

1. Rod angle at , :

   ✅

2. Slider position:

   ✅

3. Verify in the simulator. Set , , , drive the crank to , and read the slider position and rod angle. They match. ✅

4. Dead centers. The slider reaches its extremes when crank and rod are collinear: at , mm (top-dead-center), and at , mm (bottom-dead-center). The stroke is mm. These are the limit positions of the slider. ✅

Application 3: Scissor-Lift Position

Not every position problem is a four-bar loop. The scissor lift is governed by a direct geometric relationship, which makes it a clean contrast to Applications 1 and 2.

Simulator and hands-on lab

Open the Scissor Lift Simulator

Hands-on lab: Continue in the Scissor Lift Experiments lab. Experiment 1 (baseline height and force profile) plots the platform height equation derived below.

Step 1: Draw the Space Diagram and Measure

Click to reveal the graphical position solution

1. Base and arms. Draw the base. From the fixed bottom pin draw an arm of length mm at ; draw the crossing arm from the sliding bottom pin. ✅

2. Platform. Join the upper arm ends with the platform line. ✅

3. Measure. The platform height scales to mm, matching below. ✅

Step 2: Height and Spread from Geometry

Click to reveal the scissor-lift position relations

1. Each arm of length makes angle with the base. The vertical rise of one crossing is , and stacking stages multiplies it:

   ✅

2. The horizontal spread of the arm ends along the base is:

   ✅

3. Evaluate at mm, , :

   ✅

Step 3: Verify and Read the Range

Click to reveal the simulator check and limit positions

1. Verify in the simulator. Set , one stage, and drive the angle to . The reported platform height is mm. ✅

2. Limit positions. Height is maximum as (arms vertical, ) and minimum as (arms flat, ). Real lifts work in a bounded band, for example to , because the force demand near the flat position grows without bound, a result derived in the force analysis. ✅

3. Multi-stage scaling. Setting or in the simulator multiplies the height by for the same arm length, confirming . ✅

Application 4: Toggle-Clamp Limit Position

The toggle clamp is a four-bar, so its general position solution is the Freudenstein method of Application 1. Its distinctive position-analysis question is different: where is the limit (toggle) position that gives the clamp its self-locking behavior?

Simulator and hands-on lab

Open the Toggle Clamp Simulator

Hands-on lab: Continue in the Toggle Clamp Experiments lab. Experiment 1 (top-dead-centre and self-locking) locates the limit position discussed here.

Step 1: Draw the Space Diagram

Click to reveal the toggle-position construction

1. Skeleton. Draw the four-bar skeleton: ground , handle , main link , clamp arm . ✅

2. Find top-dead-centre. Rotate the handle until , and are collinear. That collinear position is the limit (toggle) position analysed below. ✅

Step 2: The Collinear (Toggle) Condition

Click to reveal the limit-position geometry

1. Limit positions of any four-bar occur when two links line up. For the toggle clamp, top-dead-center (TDC) is the handle angle at which the handle link and main link become collinear, so the toggle angle between them is zero. ✅

2. Why this is a position-analysis result. It is found purely from the geometry of the loop, with no forces involved: solve the loop closure for the handle angle that makes the two link vectors parallel. The clamp is set to rest a few degrees past this angle, the “lock margin”. ✅

3. Why it matters. At the collinear position a small handle motion produces almost no pad motion, so the velocity ratio collapses. This is the same singular configuration seen in the mobility analysis; the velocity analysis quantifies the velocity ratio while the force analysis quantifies the force amplification it creates. ✅

Step 3: Locate It in the Simulator

Click to reveal the simulator check

1. Open the simulator and drive the handle slowly. The information panel reports the TDC handle angle, the position where the handle and main link align. ✅

2. Observe the pad. As the handle approaches TDC, the pad position changes more and more slowly for the same handle increment, confirming you are at a limit position. Past TDC by the lock margin, the clamp holds itself closed. ✅

Theory: Coupler Points and Coupler Curves

A point fixed to the coupler link, but not on the line between its two joints, traces a closed path called a coupler curve as the crank rotates. Coupler curves are how a single four-bar can generate straight-line segments, figure-eights, and dwell paths used in walking mechanisms, film advances, and mixing machines.

Coupler-Point Coordinates

Let joint (crank-coupler) be at , found from the crank: , . A coupler point lies a distance from at a fixed angle from the coupler line:

Once the position analysis gives and at each instant, sweeping the crank traces the full coupler curve.

Design Guidelines for Position Analysis

Summary and Next Steps

Key Concepts Mastered

1. Vector loop method: every closed linkage gives two scalar equations per loop from the condition that the link vectors sum to zero.
2. Freudenstein equation: the four-bar position problem reduces to three constants and one quadratic, with the two roots being the open and crossed modes.
3. Slider-crank and scissor lift: their closure conditions are a single closed-form expression each, including the dead-center and height limits.
4. Grashof classification: link lengths predict whether the mechanism is a crank-rocker, double-crank, double-rocker, triple-rocker, or change-point linkage.
5. Coupler curves and limit positions: position analysis predicts the path of a coupler point and the extreme positions where the output reverses.

Position Solutions at a Glance

Mechanism	What you solve for	Closure relation	Simulator
Four-bar	,	Freudenstein quadratic	siwit.co/FBL
Slider-crank	,		siwit.co/CSM
Scissor lift	,		siwit.co/SLM
Toggle clamp	handle angle, TDC	Freudenstein, collinear limit	siwit.co/TCM

A Note on Tools

Every result in this lesson was found in closed form by hand and reproduced with a few lines of Python (NumPy). The simulators then confirm the same numbers interactively. No specialised motion-analysis software is needed; the mathematics is the tool.

Next, Velocity Analysis and Instantaneous Centers differentiates the very loop equations solved here. The four-bar position equations become velocity equations for and , and the slider-crank position gives the piston velocity, with instantaneous centers offering a second, graphical route to the same answers.

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