Four-Bar Linkage Calculator and Coupler Curve Analyzer

The position solution uses the two circle constraints |AB| = L₃ and |O₄B| = L₄. With d = O₄ − A, q = |d|, u = (L₃² − L₄² + q²)/(2q), and h = √(L₃² − u²), B = A + u d/q + σ h (−dᵧ,dₓ)/q. The branch sign σ is +1 or −1. atan2 determines each link angle in the correct quadrant, displayed in 0°…360°.

The velocity loop is L₂ω₂(−sin θ₂, cos θ₂) + L₃ω₃(−sin θ₃, cos θ₃) = L₄ω₄(−sin θ₄, cos θ₄). Its two scalar equations determine ω₃ and ω₄. Differentiating again adds the centripetal terms −ω²r and gives a second two-equation solve for α₃ and α₄. Point P then uses vP = vA + ω₃ × AP and aP = aA + α₃ × AP − ω₃² AP.

Grashof margin is (P + Q) − (S + L), using the sorted lengths. Positive means strict Grashof; zero within a relative roundoff tolerance means a change point. The shortest link's relation to the selected ground determines the inversion. Acute transmission angle is acos(|cos(θ₄ − θ₃)|). Its exact minimum follows from the endpoints of the reachable A–O₄ separation interval. Reachable input intervals are [θlow, θhigh] and [360° − θhigh, 360° − θlow]; exclude singular endpoints when solving rates.

Method sources: Williams, Ohio University ME 3011 notes, pp. 23–25 (loop closure and branches), 39–40 (Grashof), 61–63 (velocity), and 78–81 (acceleration); University of Almería, worked example 2.1 (circle construction and point kinematics). The sweep uses 241 positions at 1.5° spacing. Curve segmentation is cached independently of the playing angle.

When forming linear motion, angular derivatives are normalized by one radian, treating the radian as dimensionless in SI. The small residuals used solely for closure verification are rounded at 10⁻¹² relative length, 10⁻⁹ mm/s, and 10⁻⁸ mm/s² to suppress floating-point cancellation; computed motion retains full precision.

The drawing envelope uses only reachable sweep coordinates, the fixed pivots, and the current reachable pose. Unreachable circle-intersection constructions are excluded from both the plotted path and the viewport bounds.

• Planar, rigid, constant-length links with four ideal revolute joints; the ground is fixed.
• P is fixed to the coupler; AP and β describe a rigid offset, including an extension beyond the physical bar.
• Angular velocity and angular acceleration are instantaneous prescribed values. The angular-rate chart holds the specified ω₂ constant while varying pose.
• Crossing links may occupy parallel planes; geometric interference is not tested.
• Length inputs share a common calculation unit of mm. Formula and chart units remain as labeled when result cards are converted to US units.

Use positive link lengths and a longest link shorter than the sum of the other three. The selected angle must satisfy |L₃ − L₄| ≤ |A O₄| ≤ L₃ + L₄. A coincident A and O₄ gives no unique circle-intersection construction.

At |sin(θ₃ − θ₄)| ≤ 10⁻⁶ the velocity matrix is singular or numerically ill-conditioned, so instantaneous rate results are withheld. A warning is shown below a 5° transmission angle; this is a sensitivity notice, not an acceptance criterion. Change-point chains can jump branches and require an external constraint to select their passage through collinearity. Roundoff guards do not make an unreachable pose valid.

This is kinematics only: no forces, motor sizing, elastic deformation, joint clearance, losses, or structural safety prediction. Sweeps sample the curve and may miss a very narrow reachable interval; the reported angle limits and minimum transmission angle are analytical.