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

Methods of Kinematic Analysis

Position, velocity, and acceleration are solved by two core methods, graphical and analytical, that you then confirm with a simulator. One further trick, the instantaneous center, is worth knowing as a quick velocity shortcut, but the course does not build on it (the fourth card explains why). These are not rivals: each answers the same question in a different way, so you use them to check one another. The three analysis lessons (this one, velocity, and acceleration) apply the same set to the same mechanisms, so it is worth meeting them together before diving into the algebra.

How the methods fit together

Treat them as a sequence, not a menu: draw, solve, simulate. Draw it first with the drawing set to see the mechanism and get a quick answer at the instant of interest. Solve it analytically to turn that sketch into an exact number and, crucially, to sweep the whole cycle instead of one pose. Simulate it to animate the motion, plot a full revolution, confirm the drawing and the algebra, and drive the lab exercises. The instantaneous center sits outside this sequence as an optional one-step check on a velocity ratio, nothing more.

The real payoff is cross-checking. The length you measure off a velocity polygon should equal the value you compute by differentiating the loop, which should match what the simulator reports. When all three agree, you can trust the result; when they disagree, one of them has exposed a mistake. Every worked example in these three lessons follows this rhythm, so you build the habit of confirming a kinematic answer three independent ways.

How Close Is Close Enough?

You draw a polygon, measure mm/s, then compute mm/s. Is that a success or a mistake? Without a stated tolerance the cross-check is worthless, because any disagreement can be explained away. Here is what board work actually costs you.

Source of error	Typical size	How to reduce it
Scale rule reading	mm on the page	choose a scale that fills the page
Protractor reading		construct perpendiculars with a set square, not a protractor
Compass slip	mm	sharp lead, re-check the opening against the rule
Line thickness	mm	0.3 mm pencil, hard grade
Near-tangent intersection	unbounded	re-draw at a different instant
Combined, carefully drawn	1 to 2 percent
Combined, hurried	5 to 10 percent

The drawing is not just a warm-up for the algebra. It catches what algebra hides: a wrongly signed still produces a plausible number, but it produces a visibly wrong polygon. A sliver of a triangle warns you the mechanism is near a singularity long before you notice a small denominator in a formula. And the arcs, tangents and perpendiculars you strike by hand are the same geometric constraints you will later apply in a parametric CAD sketch.

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

The graphical method solves the position problem with the drawing set alone, no equations. It is the fastest way to see the answer, and it doubles as a check on the algebra that follows. The whole trick is that two of the four link lengths ( and ) are fixed, so the joint must lie at a known distance from both and , and the only such points are where two arcs cross.

Click to reveal the graphical position solution (arc-intersection)

1. Choose and note a scale. Pick a length scale that fills the page, say 1 cm = 20 mm, and write it on the drawing. Every length below is laid off with the scale rule at this scale. ✅

2. Ground and crank. Draw the ground mm horizontally. With a protractor set at and lay off the crank mm with the scale rule. Point is the crank pin. ✅

3. Intersect the coupler and follower arcs. Open the compass to the coupler length mm and strike an arc centred on ; reset it to the follower length mm and strike an arc centred on . The arcs cross at two points, the two assembly modes. Take the upper crossing (the open mode) as the coupler-follower joint ; the lower crossing is the crossed mode. ✅

4. Measure the unknowns. Draw (coupler) and (follower), then read their angles from the horizontal with the protractor: and . ✅

The drawing gives each angle to within a degree or so. The closed-form Freudenstein solution below pins them to and , and the simulator confirms both. Reading to about is normal for board work, enough to catch a gross error and to tell the two assembly modes apart, which is exactly what the graphical method is for.

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. Substitute back into the original loop equation. This is the mandatory last step of every position solution, not a flourish (, , ):

   ✅

   Both residuals should come out to a fraction of a millimetre against link lengths of tens of millimetres. If either is a few millimetres or more, something is wrong: most often a mismatched assembly mode, an angle left in degrees where radians were needed, or a sign slip in a quadratic coefficient. The loop either closes or it does not, and this check takes fifteen seconds. Make it a habit now and it will save an entire question later. ✅

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

The graphical method locates the piston with the drawing set alone. The crank pin rides a circle of radius , and the piston lies on the centre-line exactly one rod length from , so swinging the rod from onto the centre-line fixes it.

Click to reveal the graphical position solution (rod-swing)

1. Choose and note a scale. Pick a scale that fills the page, say 1 cm = 20 mm, and mark it on the drawing. ✅

2. Centre-line and crank. Draw the slider centre-line through . With the protractor set at and lay off the crank mm with the scale rule; is the crank pin. ✅

3. Swing the rod. Open the compass to the rod length mm, centre it on , and cut the centre-line at , the piston. ✅

4. Measure. Scale the piston distance mm, and read the rod angle from the centre-line with the protractor. ✅

The drawing gives and to within a millimetre and a degree. The loop equations below give the exact and , and the simulator confirms both.

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

The graphical method reads the platform height straight off a scaled drawing of one scissor stage. Each arm pivots at its centre, so the platform rises as the arms swing up from the horizontal.

Click to reveal the graphical position solution

1. Choose and note a scale. A scissor arm is long, so pick a scale such as 1 cm = 40 mm and mark it on the drawing. ✅

2. Base and arms. Draw the base line. With the protractor set , lay off one arm of length mm from the fixed bottom pin with the scale rule, and draw the crossing arm from the sliding bottom pin at the same angle. ✅

3. Platform. Join the two upper arm ends with the platform line, parallel to the base. ✅

4. Measure. Scale the vertical gap between the base and platform: mm, matching below. ✅

The measured height confirms the closed-form solved next, and the simulator confirms both.

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

The graphical method finds the toggle (limit) position by construction: the clamp reaches its limit when the handle and the main link line up, so you swing the handle until the drawing shows that alignment.

Click to reveal the toggle-position construction

1. Choose a scale and draw the skeleton. Mark a scale (say 1 cm = 20 mm) and lay off the four-bar skeleton to scale with the scale rule and compass: ground , handle , main link , clamp arm . ✅

2. Find top-dead-centre. With the compass, swing the handle until , and fall on one straight line (check the alignment with the straight edge of a set square). That collinear position is the limit (toggle) position analysed below. ✅

3. Measure. Read the handle angle at the toggle position with the protractor; it fixes the clamping geometry solved for exactly next. ✅

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.

Application 5: Quick-Return Position (a Slider-Crank Inversion)

The metal shaper cuts a flat surface by driving a single-point tool back and forth across the work while the table steps sideways. Cutting happens on the forward stroke only, so the ram should move slowly and steadily as it cuts and then snap back quickly on the idle return. The drive that does this is the crank-and-slotted-lever quick-return mechanism, and it is the same four-link slider-crank chain as the engine of Application 2 with a different link held fixed: fix the frame and you get the reciprocating engine, fix the link carrying the sliding block and you get the shaper’s quick return. The geometry alone sets how much faster the return is than the cut.

Hands-on lab

Open the Crank-Slider Simulator

Hands-on lab: This inversion has no simulator of its own, but the quick-return principle is Experiment 2 of the Crank-Slider Experiments lab: add an offset and the forward and return strokes become unequal, the same asymmetry the shaper drives to an extreme. The shaper numbers below come from the drawing and the trigonometry, which confirm each other.

Step 1: Draw the Space Diagram and Measure

The graphical method settles this mechanism with the drawing set alone, because the two extreme positions of the lever are simply the two tangents drawn from the fulcrum to the crank circle. At an extreme, the sliding block sits at the tangent point, so the crank is perpendicular to the lever .

Click to reveal the graphical position solution (tangent extremes)

1. Choose and note a scale. Pick a scale that fills the page, say 1 cm = 40 mm, and mark it on the drawing. ✅

2. Fixed centres. Mark the fulcrum , and directly above it the crank centre with mm to scale. The line is the centre-line the lever swings about. ✅

3. Crank circle. Open the compass to mm and draw the crank circle about . ✅

4. Extreme lever positions. Draw the two tangent lines from to this circle; they touch at and . Extend each to the driving point at mm to get the ram-end extremes and . These two lines are the lever at the ends of its swing. ✅

5. Measure. Read the half-swing each side of (total swing about ) with the protractor; the crank angle between the extremes on the side measures about and on the far side about ; the ram travel mm with the scale rule. ✅

The drawing gives the swing, the crank angles, and the stroke to within a degree and a millimetre. The trigonometry below gives them exactly, and they agree.

Step 2: Solve the Geometry

Click to reveal the closed-form quick-return solution

1. Half-swing of the lever. At each extreme the crank is perpendicular to the lever, so triangle is right-angled at the tangent point :

   The lever swings in total. ✅

2. Crank angles for the two strokes. Because the crank is perpendicular to the lever at both extremes, the crank angle swept while the ram makes its return (the arc nearer ) is

   ✅

3. Time ratio. The crank turns at constant speed, so the stroke times are in the ratio of these angles:

   The return stroke is about times faster than the cut. ✅

4. Ram stroke. The driving point swings through about at radius , so the stroke is the chord between its extremes:

   ✅

Step 3: Confirm the Principle

Click to reveal the drawing-versus-calculation check and the simulator principle

1. Drawing against calculation. The construction measured about and for the crank angles and mm for the stroke; the trigonometry gives , , and mm. They agree to within drawing tolerance, which is the confirmation this inversion needs. ✅

2. The principle in the simulator. Open the crank-slider simulator and choose the Quick-Return (offset) preset (crank , rod , offset mm). The forward and return strokes take unequal times, a time ratio of about : the same quick-return asymmetry, in its milder offset-slider form. The shaper’s crank-and-slotted-lever drives that asymmetry harder still by placing the fulcrum close to the crank. ✅

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
Quick-return (shaper)	time ratio, stroke	, stroke	drawing calc

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, drawn first as a velocity polygon and then confirmed by the differentiated loop.

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