Delta Robot Design — 3-DOF Parallel Robot Mechanical Design

Mechanical design project — Ho Chi Minh City University of Technology and Education (HCMUTE). Complete mechanical design of a ceiling-mounted 3-DOF Delta parallel robot for high-speed pick-and-place, benchmarked against the ABB IRB 360 FlexPicker.

This repository contains the SolidWorks CAD model, the MATLAB analysis and simulation code, the finite-element (FEA) results, the machining drawings, and the electrical / control-system and machine-vision design. Every conclusion is backed by data read back from the model or the solver — no assumed figures. All figures below are taken from the design analysis, the FEA results and the control-system design.

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Overview

Architecture	3-DOF Delta (parallel), rotational actuators, ceiling-mounted
Payload	2 kg
Workspace	cylinder Ø800 × 250 mm, centred ≈ 925 mm below the shoulder plane
Pick-and-place cycle	1.2 s (quintic trajectory)
Peak TCP speed / acceleration	≈ 1875 mm/s / ≈ 1.18 g
Actuator	Wittenstein TPMA010S-055T gearmotor, ratio i = 55
Structural material	Aluminium 6061-T6 (all machined parts)
Total mass	122.4 kg (CAD model), ≈ 140 kg estimated real
Minimum factor of safety (whole robot, full workspace)	7.1

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1. Kinematic architecture

Symbol	Meaning	Value
L₁	upper-arm (bicep) length: shoulder axis → elbow ball line	407.5 mm
L₂	forearm length, ball-centre to ball-centre	1000.0 mm
R	base joint-circle radius	347.0 mm
r	moving-platform radius	120.6 mm
R − r	base radius − moving-platform radius	226.4 mm
φᵢ	arm azimuths	−90°, 30°, 150°
θᵢ	motor-angle range (elbow-down)	−45° … 70°

Fig. 2.1 / 2.3. Basic structure of the three-arm RUU Delta robot (left); CAD model and the general geometric parameters R, r, L, b, H (right).

Geometric model

Each arm $i$ has a fixed azimuth $\phi_i$; the three arms sit at −90°, 30° and 150°. Writing $c_i=\cos\phi_i$ and $s_i=\sin\phi_i$, and measuring the motor angle $\theta_i$ downward from the shoulder plane, the base joint, the platform joint and the elbow of arm $i$ are

$$\mathbf{B}_i = R,(c_i,\ s_i,\ 0)$$

$$\mathbf{P}_i = \mathbf{P} + r,(c_i,\ s_i,\ 0)$$

$$\mathbf{E}_i = \mathbf{B}_i + L_1,(c_i\cos\theta_i,\ s_i\cos\theta_i,\ -\sin\theta_i)$$

The rigid forearm gives one closure equation per arm:

$$\lVert \mathbf{E}_i - \mathbf{P}_i \rVert = L_2$$

Inverse kinematics (closed form)

For a target $\mathbf{P}=(x,y,z)$ the closure equation of each arm reduces to a single equation in $\theta_i$:

$$E_i\sin\theta_i + F_i\cos\theta_i + G_i = 0$$

$$u_i = x,c_i + y,s_i - (R-r), \qquad w_i = -x,s_i + y,c_i$$

$$E_i = 2 L_1 z, \qquad F_i = -2 L_1 u_i, \qquad G_i = L_1^2 + u_i^2 + w_i^2 + z^2 - L_2^2$$

With $\rho_i=\sqrt{E_i^2+F_i^2}$ and $\psi_i=\mathrm{atan2}(F_i,\ E_i)$:

$$\theta_i = \arcsin!\left(\frac{-G_i}{\rho_i}\right) - \psi_i$$

taking the elbow-down root inside the range −45° … 70°. A real solution exists only if $|G_i| \le \rho_i$; otherwise $\mathbf{P}$ lies outside the reachable workspace.

Forward kinematics

Given the three angles, shift each elbow to a virtual centre $\mathbf{C}_i = \mathbf{E}_i - r,(c_i,\ s_i,\ 0)$. The platform centre $\mathbf{P}$ is the intersection (trilateration) of three spheres of radius $L_2$ centred at $\mathbf{C}_1,\mathbf{C}_2,\mathbf{C}_3$, keeping the lower-$z$ root. Round-trip check: the error $\lVert \mathbf{P} - \mathrm{FK}(\mathrm{IK}(\mathbf{P})) \rVert$ stays below $6.2\times10^{-13}$ mm over 3000 points (see §2).

Velocity Jacobian and singularity

Differentiating the three closure equations gives

$$\mathbf{A},\dot{\mathbf{P}} = \mathbf{B},\dot{\boldsymbol{\theta}}, \qquad \dot{\mathbf{P}} = \mathbf{J},\dot{\boldsymbol{\theta}}, \qquad \mathbf{J} = \mathbf{A}^{-1}\mathbf{B}$$

Row $i$ of $\mathbf{A}$ is the forearm direction $(\mathbf{E}_i-\mathbf{P}_i)^{T}$, and $\mathbf{B}$ is diagonal with

$$B_{ii} = L_1,(\mathbf{E}_i-\mathbf{P}_i)^{T},\hat{\mathbf{v}}_i, \qquad \hat{\mathbf{v}}_i = (-c_i\sin\theta_i,\ -s_i\sin\theta_i,\ -\cos\theta_i)$$

Here $\det\mathbf{A}=0$ marks a parallel (Type II) singularity, $\det\mathbf{B}=0$ a boundary (Type I) singularity, and the condition number $\kappa(\mathbf{J})$ measures how close a pose is to one. Over the target workspace $\kappa(\mathbf{J}) \le 2.75$ with no singular points.

2. Kinematics verification

The inverse and forward kinematics were implemented independently and cross-checked:

• Round-trip error ‖P − FK(IK(P))‖ ≤ 6.2 × 10⁻¹³ mm over 3000 sample points.
• Reachable radius 650 mm at z = −925 mm (requirement: 400 mm).
• Jacobian: no singular points inside the workspace; condition number κ(J) max 2.75, average 2.13; minimum forearm–bicep transmission angle 49.8°.

Fig. 2.8 / 3.6. Simulation check of the kinematic computation (left); distribution of the Jacobian condition number and the force-transmission angle over the working region (right).

Fig. 3.4 / 3.2. Workspace point cloud from the three joint-angle sweep (left); workspace map at y = 0 with the pick-and-place trajectory check (right).

3. Motion & trajectory

The pick-and-place cycle is planned as a fifth-order (quintic) polynomial and the joint kinematics are checked against their bounds.

Fig. 3.8 / 3.9. Pick-and-place trajectory interpolated with a fifth-order polynomial (left); joint angle, angular-velocity and angular-acceleration bounds along the trajectory (right). Peak joint speed 30.1 rpm.

4. Force & dynamic analysis

Forearm links are solved as two-force members; joint torques are obtained through the Jacobian. Loads are swept over the whole workspace (centre + 24 edge points + 4 waypoints × 26 peak-acceleration directions) and the worst case is retained.

Quantity	Nominal (centre pose)	Worst case (workspace edge, R400 / z−1050)
End-effector force F_ee	175.8 N	176.1 N
Forearm-pair force	122.7 N (61.3 N per rod)	182.4 N (× 1.49)
Bicep bending moment	≈ 50 N·m	74.3 N·m (× 1.49)
Platform joint torque τ	48.4 N·m	—
Jacobian condition number	—	2.75 (no singularity)

Fig. 4.2 / 4.3. Force diagram at the moving platform (left) and in the passive (long) arm (right).

Fig. 4.4 / 4.5. Force diagram of the active arm (left); torque at the active joint versus tilt angle α (right).

5. Actuator selection

The required joint torque is split into static and dynamic terms and multiplied by a 1.5 safety factor (per the advisor's method), including the bicep weight and inertia (6.9 kg, from CAD) and the reflected rotor inertia J₍mot₎·i²:

Criterion	Result	Margin
Required torque M_yc = (M_static 35.8 + M_dynamic 55.0) × 1.5	136.3 N·m ≤ T2B 230 N·m	1.69 ×
RMS torque M_rms	53.4 N·m ≤ 110 N·m	2.06 ×
Joint speed	30.1 rpm ≤ 88 rpm	pass

The initially considered TPM-010S-061T (T2B 80 N·m) is rejected — M_yc 112.5 N·m > 80 N·m. The high-torque TPMA010S-055T passes all three criteria.

Fig. 4.6. Selected gearmotor with reduction unit — Wittenstein TPMA010S-055T.

6. Structure & materials

• All machined parts: aluminium 6061-T6. The base was first designed in ASTM A36 steel but was too heavy, so all three base blocks were reassigned to 6061-T6 (−128.8 kg).
• Forearm connecting rods: carbon fibre. Ball-joint rod ends (60645K471): alloy steel.
• The base plate (DR-001) is split into three separate fabrication blocks — mounting plate, welded frame, and hanging lid — bolted/welded together.
• Part numbering: DR-000 main assembly, DR-001…DR-007 machined parts; purchased parts keep their vendor names.

Fig. 4.1. 3D design model in SolidWorks.

7. Vacuum gripper (end effector)

The end effector is a vacuum gripper at the platform centre, facing down, that picks parts off the conveyor and sorts them by shape (triangle / circle / square). The cup, valve, tubing and sensor mass is not part of the 2 kg payload — it is added to the dynamic model once the hardware is fixed; the grip is sized against the 2 kg payload alone.

Required suction force (weight + lift inertia, with a pull-off safety factor):

$$F_s = k,m,(g + a_z)$$

Total cup area and minimum working vacuum for $n$ round cups of diameter $d$ and effective efficiency $\eta$:

$$S = n,\pi,\frac{d^{2}}{4}, \qquad p_\text{req} = \frac{F_s}{\eta,S}, \qquad F_\text{cap} = \eta,p_\text{max},S \ge F_s$$

With $k = 2.0$, $m = 2.0$ kg, $g = 9.81$, $a_z = 11.58$ m/s², $n = 2$ cups, $d = 40$ mm, $\eta = 0.70$:

Quantity	Value
Required suction force $F_s$	85.56 N
Total cup area $S$	2.513 × 10⁻³ m²
Minimum vacuum $p_\text{req}$	48.7 kPa
Holding force at $p_\text{max}=60$ kPa	105.6 N ≥ 85.56 N ✓

Preliminary cup layout by shape: triangle 2 × Ø40 (or 4 × Ø30), circle 2 × Ø40 symmetric about the centre, square 4 × Ø30.

Fig. 2.10. Forces on a part lifted vertically by the vacuum cup: suction force F_s up, weight + lift inertia F = F_g + F_a down.

8. Structural validation (FEA)

SolidWorks Simulation, static studies on the load-bearing parts with the peak dynamic loads from §4, including a mesh-convergence sweep and a full-workspace multi-pose sweep.

Part	von Mises (worst pose)	Factor of safety
DR-006 Elbow clevis	38.8 MPa	7.1 (governing)
DR-005-2 Upper-arm link	—	43.5
DR-007 Moving platform	—	14.7
DR-001-1 Base plate (aluminium)	—	≈ 1268
DR-001-3 Ceiling lid (aluminium)	—	≈ 72

Fig. 4.7 / 4.8. von Mises stress — DR-001-3, DR-001-2 and DR-001-1 (left); DR-006, DR-007 and connecting rod 6516K305 (right). The whole-robot minimum factor of safety over the entire workspace is 7.1.

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9. Electrical & control system

The control system is a Siemens motion-control stack. An industrial PC (SIMATIC IPC) runs the vision pipeline and streams pick coordinates over PROFINET to a SIMOTION D motion controller, which carries a built-in Delta technology object (synchronous three-axis interpolation). SIMOTION drives three SINAMICS S120 motor modules over DRIVE-CLiQ, one per arm, each powering a TPMA010S-055T gearmotor with incremental-encoder feedback. The three motor modules share one Active Line Module (common DC bus), so braking energy is recovered between axes.

Safety is a hardwired chain independent of the main controller: an E-Stop / limit-switch loop (Siemens 3RG40 / SIRIUS 3SK) triggers STO (Safe Torque Off, IEC 61800-5-2) directly on the SINAMICS drives through a SIRIUS 3SK safety relay.

Servo power is 320 V AC (per the TPMA010S-055T datasheet); SIMOTION, the SIRIUS relay and the IPC share a 24 V DC control supply (IEC 61131-2); the camera is powered over its GigE link. DC-supply and fuse ratings, and the conveyor branch, are not yet dimensioned.

Link	Bus / standard	Purpose
Camera (Basler acA1300-30gc) ↔ SIMATIC IPC	GigE Vision (Ethernet ≤ 1000 Mbps)	raw high-speed image transfer
SIMATIC IPC ↔ SIMOTION D	PROFINET IO (real-time industrial Ethernet)	pick coordinates / status
SIMOTION D ↔ SINAMICS S120 (×3)	DRIVE-CLiQ (internal bus, same rack)	servo control, real-time encoder feedback
SIRIUS 3SK ↔ SINAMICS S120	STO (IEC 61800-5-2, hardwired)	controller-independent safe torque off
E-Stop / limit switch (3RG40) ↔ SIRIUS 3SK	hardwired digital I/O	safety signal
SINAMICS S120 ↔ TPMA010S-055T (×3)	power cable + incremental-encoder cable	servo power, position feedback
AC mains ↔ SINAMICS S120	320 V AC	drive power
DC supply ↔ IPC / SIMOTION / SIRIUS	24 V DC	control power

Fig. 5.6. Control-system block diagram — vision IPC, SIMOTION D motion controller, three SINAMICS S120 drives on a shared DC bus, and the SIRIUS / STO safety branch.

Fig. 5.7. Power distribution — AC mains through E-Stop and fuse to 320 V AC for the three drives and a 24 V DC control supply for the IPC, SIMOTION D and SIRIUS relay.

10. Machine vision (image processing)

Each frame is converted from BGR to HSV, then two branches run in parallel. The shape branch performs corner / blob detection and template matching, classifying every object as circle, square or triangle. The colour branch applies fixed HSV thresholds for red, green and yellow and detects blobs by area. The two results are merged into a (colour, shape) label per object, and the corresponding pick coordinate is sent to the IPC.

Colour	Lower (H, S, V)	Upper (H, S, V)
Red	(173, 0, 0)	(255, 255, 255)
Yellow	(18, 0, 76)	(36, 255, 255)
Green	(58, 155, 114)	(85, 255, 255)

On a synthetic test image of 9 objects (3 shapes × 3 colours) on a dark, conveyor-like background, the algorithm returned the correct (colour, shape) label for all 9, and the red threshold isolated a clean binary mask containing only the 3 red objects — confirming the pipeline before integration with a real camera.

A pixel coordinate (u, v) is mapped to a physical pick point (X, Y) through three coordinate frames — camera {C}, robot base {B} and conveyor {W} — by a linear calibration.

Fig. 5.1. Colour- and shape-recognition algorithm: parallel shape branch (corner / blob detection, template match → circle / square / triangle) and colour branch (HSV threshold → red / green / yellow, blob by area).

Fig. 5.5. Threshold tuning per colour (left) and the resulting multi-object classification on the simulated conveyor image (right).

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11. Design report — chapters

Chapter	Title	What it covers
1	Overview	Motivation; state of the art of Delta robots; general and specific objectives; research object and scope; research method; report outline.
2	Theoretical basis	Delta structure and working principle; geometric model and coordinate frames; kinematic foundations — degrees of freedom, geometric parameters, per-limb closed-loop equations, forward kinematics, inverse kinematics, model-consistency check; basis for the vacuum-gripper design — joint layout and suction-cup arrangement, suction force and vacuum pressure.
3	Workspace computation & simulation	Objectives, scope and input data; the L9 survey parameter set and the test trajectory; point-density workspace simulation — joint-space sweep, volume and largest-cross-section estimate; verification of the target Ø800 × 250 mm cylinder, of the pick-and-place trajectory, and of the joint velocity / acceleration bounds.
4	Load analysis, material & drivetrain selection	Material selection; part masses from CAD; load data and force-distribution model; internal-force analysis of the links — free-body model, axial forces, motor–gearbox selection; FEA strength check of the main load-bearing parts; overall strength summary versus requirements.
5	Electrical & control system design	Machine-vision system — colour/shape recognition algorithm and test results on sample images; camera–robot–conveyor coordinate transforms; control-board architecture.
6	Conclusion & future work	Results, limitations of the work, and development directions.

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_{The full design report, third-party vendor CAD and reference papers are not included in this repository (see .gitignore). All rights reserved; please cite if reused. © JohnNguyen205, HCMUTE.}