# Introduction

The delta robot is a type of parallel robot introduced by Professor Reymond Clavel in 1986. The motors are fixed to ground and the moving parts are kept light as possible in order to have a high robot velocity and acceleration (e.g. 15g in industrial environments). However  the ratio between payload and robot mass is lower (payload is limited to 5÷10kg) and the repeatability is worse than other robot types (e.g.  anthropomorphic robot and SCARA). The delta robot is ideal for use in food and pharmaceutical industries and in packaging tasks thanks to its high velocity.

## Robot architecture

The delta robot is made by 3 arms that connect the end effector to ground by 3 parallelograms. The end effector can only move in 3 directions (x, y and z) without rotating, then the DOF number is 3. Each arm is made by following links:

• a rocker arm (or upper link) connected to motors;
• a pair of connecting rods (or lower links) connected to the rocker arm and the end effector by some rod ends.

The rotational motion of rocker arm is controlled by motor while the rod ends are passive joints. Two versions were developed:

• 3 axes version: the end effector can only translate without rotating;
• 4 axes version: the end effector can translate and rotate around a vertical axis thanks to an another motor.

The analyzed version is the 3 axes one.

# Objectives

The analysis objectives are the following:

• Define a method to identify and analyze the robot workspace. The developed method is used to analyze the effect of geometric parameters on robot workspace. Then the velocity ratios between active joint and end effector motion are analyzed and a scalar index is introduced in order to identify the regions close to singularity configurations.
• Analyze the connecting rods flexibility influence to the robot dynamics.  First the vibration modes are identified, then the circular motion response and the Adept cycle one are obtained.

# The modelling problem

The mechanical system is a closed kinematic chain made by N=10 bodies (escluding ground) connected by revolute and spherical joints:

• 3 rocker arms that are connected to ground by revolute joints (active joints). The rotational motion θ_i of i-th rocker is driven by motor and limited between θ_max and θ_min.
• 3 pairs of connecting rods that connect the rocker arms to the end effector by spherical joints (passive joints).
• End effector.

There are R=3 revolute joints and S=12 spherical joints, then the Grubled count provides:

k=6N-6F-5T-4H-3S-3P=6⋅10-5⋅3-3⋅12=9 DOFs

3 DOFs are the 3 driven axes (i.e. rocker arms rotations) and 6 DOFs are the spin motions of connecting rods around their axes. The last ones are stopped by some links or springs that connect each pair of conneting rods. Kinematics of the mechanical system and available workspace provided by industrial robot manufacturers

## Analytical model

An analytical model is developed to understand which geometric and physical parameters influence the robot kinematics and dynamics.

### Kinematic model

The constraint equations of the mechanical system are computed with following assumptions:

• the end effector can only translate (i.e. α_e=β_e=γ_e=0);
• each pair of connecting rods is parallel (i.e. α_ia=α_ib=α_i and β_ia=β_ib=β_i).

A frame and a 4×4 transformation matrix are defined for each body. The connecting rod orientation is determined by Tait-Bryan angles α_i and  β_i to avoid singularity conditions and to constraint their spin motion. The costraint equation for a generical spherical joint between bodies i and j in the point P is: where 0 is the ground frame. The constraint equations of i-th arm are obtained applying this equation for each spherical joint: where The robot kinematics is defined by 5 geometrical parameters: l_1, l_2, l_4, l_5 and l_6. There are m=9 constraint equations f(q) and n=12 generalized coordinates q, so the DOF number is k=n-m=3.

### Dynamic model

The II order DAE system of a mechanical system is where the mass matrix M and the force vector F are obtained by Lagrange equation. The kinetic and potential energies of the mechanical system are  The robot dynamics is defined by 10 physical parameters: rocker mass m_r, rocker CG position xG_r and yG_r, rocker inertia Ir_zz, connecting rod mass m_c,  connecting rod  CG position zG_c, connecting rod inertia matrix Ic_xx, Ic_yy and Ic_zz and end effector/payload mass m_e.

The ADAMS model of the robot is fully parameterized by the showed geometrical and physical parameters.  A design variable is created  for each parameter and the default value is set according to following table. The values are estimated from CAD model of Omron’s Adept Hornet 565 robot  and its technical manual . The used measure unit system is MKS. Default values of ADAMS model design variables.

Each robot link is modelled as a cylinder with proper length and radius to have a good graphical view (these dimensions are not important from kinematic and dynamic point of view). The mass properties of bodies are setted according to defined physical parameters using the design variables. Other mass properties are set to 0m  (if length) or 1e-3 kgm^2 (if moment of inertia). The joints are:

• revolute joint between rocker arm and ground;
• spherical joint between rocker arm and connecting rod;
• spherical joint between connecting rod and end effector.

The joints markers are defined using the ADAMS function LOC_RELATIVE_TO({x,y,z}, MARKER_N) to fully parameterize the model by the design variables. Other 6 general constraints GCON are introduced in order to constraint the connecting rods spin motions with the following expression:

PHI(MARKER_N)=0

where MARKER_N is the marker that identified each connecting rod orientation. The model verify provides

VERIFY MODEL: .delta_robot
9 Gruebler Count (approximate degrees of freedom)
10 Moving Parts (not including ground)
3 Revolute Joints
12 Spherical Joints
3 Degrees of Freedom for .delta_robot
There are no redundant constraint equations.

where the Grubler count is 9 because the GCONs are not considered while the DOF number is 3. The motions that controlled the DOFs are the following:

• 3 rotational joint motions activated when dinverse dynamic analysis is performed;
• a general point motion to the end effector activated when inverse kinematic analysis is performed.

The ADAMS model is verified performing 2 motions:

• circular motion.

The Adept cycle is a standard symmetric cycle 25/305/25 used by Omron’s Adept to analyze the robot performances. The path is made by a vertical motion of h=25mm (depart), a horizontal motion of l=305mm (travel) and a vertical motion of h=25mm (approach). The cycle time is T_c=0.37s with a 1kg payload , so the simulated motion time is T=T_c/2=0.185s. The chosen start point is x=152.5mm, y=0mm and z=700m and the horizontal motion is along x direction.  The used motion law is a velocity trapeze with a maximum acceleration about 7g according  to  robot manual  and the cartesian trajectory is defined by via-points function. Adept cycle motion laws: x (a), y (b) and z (c). Reference values are T_c=0.37s, l=305mm, h=25mm and z0 = 700mm.

First, the end effector motion is controlled by the general point motion and an inverse kinematic analysis is performed in order to determine the revolute joints rotations θ_1(t), θ_2(t) and θ_3(t).

The revolute joints rotations θ_1(t), θ_2(t) and θ_3(t) are set to the rotational joint motions and an inverse dynamic analysis is performed in order to determine the active joints torques and powers. The maximum mechanical power is about 920W according to robot manual value about 1160W  (friction and other power losses are not considered).

### Circular motion

The circular motion is on xy plane at z=700mm. The radius R=100mm and velocity V=2.6m/s are set to have a lateral acceleration A_lat = V^2/R=6.9g according to robot manual . The motions law is the following: First, an inverse kinematic analysis is performed in order to determine the revolute joints rotations θ_1(t), θ_2(t) and θ_3(t). The end effector orientation (Tait-Bryan angles) don’t change, so the end effector only moves along x, y and z directions without rotate. Revolute joints rotations in the circular motion. Reference values are R=100mm and V=2.6m/s. End effector orientation (Tait-Bryan angles) in the circular motion.

The revolute joints rotations θ_1(t), θ_2(t) and θ_3(t) are set to the rotational joint motions and an inverse dynamic analysis is performed in order to determine the active joints torques. The torques are cyclically symmetric, so the robot model symmetry is verified.

# Simulation and analysis of results

## Workspace analysis

A simple method to determine the robot workspace is the brute force search, i.e. the end effector position is determined for each combination of θ_1, θ_2 and θ_3 in the range [θ_min, θ_max]. However, there are many combinations if also the geometrical parameters are changed to analyze their influence (e.g. there are 10^5=100000 combinations with a 10 points discretization and 2 geometrical parameters). The brute force search is used in order to obtain the workspace geometry of the Omron’s Adept Hornet 565 robot and to define a general identification method of the robot workspace. The passive joints limits are not considered in these analysis.

### Robot workspace geometry

The workspace geometry of  Omron’s Adept Hornet 565 robot is determined by the brute force search with θ_min=-109° and θ_max=51° . A DOE is performed using 10 levels for each revolute joint rotation (i.e. 10^3=1000 combinations). The workspace have a radial symmetry and its boundary surfaces are:

• upper boundary surface obtained with θ_i = θ_max;
• lower boundary surface obtained with θ_i = θ_min. Workspace of the Omron’s Adept Hornet 565 robot. In figure (c) the available workspace provided by Omron  is shown.

### Workspace identification method

The boundary surfaces is partially defined by 2 geometrical dimensions:

• maximum radius r (horizontal extension);
• height h (vertical extension).

These dimensions are determined by points P_1, P_2 and P_3. The point P_1 is obtained when each revolute joint angle is θ_i = θ* where θ* is:  Workspace boundary on xz plane (a) and maximum z configuration (b).

The 2 dimensions are obtained by a robot trajectory through P_1, P_2 and P_3. The motion law is: where the motion time T is arbitrary (e.g. T=1s).

The values of  r and h are obtained as: where z_e(0) is determined by the following expression:

IF(time : 0, z_e , 0 )

### Geometrical parameters influence

The developed method is used in order to analyze the influence on r and h of 2 geometrical parameters:

• rocker arm length l_2;
• connecting rod length l_4.

A DOE is performed using 32 levels for each geometrical parameters (i.e. 32^2=1024 combinations) and a variation about 30%. Other geometrical parameters are set to their default values. The results show that:

• h depend on l_2 while the parameter l_4 shifts the workspace vertically;
• r depend on l_4. h and r isocurves. Reference values are l_20=315mm, l_40=825mm, h_0=627mm and r_0=785.1mm

4 configurations A, B, C and D are analyzed through brute force search in order to confirm the obtained results. Configurations A, B, C and D. Subscript “r” indicates the value obtained through brute force search and subscript “m” indicates the value obtained through develped method.

### Velocity ratios analysis

The velocity ratios between the end effector motion and active joints rotations are important from kinematic and dynamic points of view:

• Singularity configurations are defined by velocity ratios.
• External forces on the end effector are transmitted to joints through velocity ratios. The joints torques are related to end effector force by the following equation: where r_v is the velocity ratio matrix, F_r is the end effector force vector and τ is the joints torques vector.

• Reduced inertia of payload depend on velocity ratios. Considering only the end effector mass m_e the EOMs are: where v=[θ_1, θ_2, θ_3]^T, so m_e⋅r_v^T⋅r_v is the reduced inertia matrix.

A static analysis is performed in order to determine the velocity ratios. A unit force is applied along each direction and the velocity ratios matrix is determined by following equation: A scalar index k is introduced in order to evaluate singularity configurations and reduced inertia variation • if k is close to 0 then the region is near to a singularity configuration;
• if k is variable then there is a variation of reduced inertia.

The workspace boundary provided by Omron avoids the regions where k is variable or close to 0. Scalar index as a fuction of end effector position in the Omron’s Adept Hornet 565 robot.

The scalar index k is analyzed in the 4 configuration A, B, C and D:

• In configuration A the scalar index is high because dimensions l_2 and l_4 are high, so the requested joints torques are greater. However the variation is low and the singularity regions are small.
• In the configuration B there is an high variation and the  singularity regions are wide, so the available workspace is very reduced.

The singularity configurations don’t increase if parameters l_2 and l_4 are both increased but only if a single parameter is modified. However the reduced inertia increases if parameters l_2 and l_4 are both increased because the force arm is greater. Scalar index k as a funtion of end effector position in the 4 configurations A, B, C and D.

## Conneting rods flexibility analysis

In delta robots the moving parts are kept lights as possible to have ahigh robot velocity and acceleration. However the links are more flexible and the robot is more sensitive to vibration, so the moving time is greater and the precision is worse. A FEA model of the connecting rod is included in ADAMS model through CMS technique in order to analyze the effect of connecting rod flexibility on robot dynamics.

### Analytical and FEA model of the connecting rod

The connecting rod is modelled using 10 beam elements through NASTRAN. The mechanical parameters are:

• mass per unit length μ=104e-3kg/m;
• elastic modulus and section inertia product EI=174Nm^2.

The values are estimated by robot CAD model and some technical manuals. The first 4 natural frequencies of  the FEA model are compared to an analytical model in the cases of simply supported beam and double fixed beam. The analytical natural frequencies are computed as where β_k is computed by:

• Simply supported beam • Double fixed beam ADAMS applies the following damping ratios by default:

• ζ=0.01 for f<100Hz;
• ζ=0.10 for f=100÷1000Hz;
• ζ=1 for f>1000Hz.

The vibration modes greater than 1000Hz don’t vibrate, so only the natural frequencies lower than 1000Hz are compared. The difference between analytical natural frequencies and FEA model ones is lower than 3%, so 10 beam elements are sufficient. Comparison between natural frequencies obtained by analytical model and by FEM/ADAMS model.

### Vibration modes

The FEA model of the connecting rod is introduced in the ADAMS model. The robot vibration modes are determined in the configuration θ_1=θ_2=θ_3=0.

Two types of vibration modes are identified:

•  modes with a variation of end effector orientation (e.g. modes 13 and 14);
• modes without a variation of end effector orientation (e.g. mode 3).

The end effector rotates as a consequence of connecting rods flexibility, so the gripper vibrates along and around x and y directions. Modes of vibration with (a and c) and without (b and d) variation of end effector orientation.

### Circular motion response

The steady-state response of circular motion is obtained with a radius R=100mm and a velocity V=2.6m/s in order to have a lateral acceleration A_lat = V^2/R=6.9g. Before perform the analysis, the natural frequencies of 13th and 14th modes are computed as a function of circular path position φ=ω⋅t. Natural frequencies variation of 13th and 14th modes ad a function of circular path position.

The dynamic analysis is performed for 10 periods in order to overcome the initial transitory and to obtain the steady-state response for a sufficient time. The variations of end effector position (Δx and Δy) and orientation (Δα and Δβ) are measured in a rotating frame with an angular velocity ω=V/R=26rad/s. There is a initial transitory until 0.25s in which the system vibrates with the13th and 14th natural frequencies. In steady-state conditions the vibration frequency is 3 times the rotating frequency and the excitation is related to natural frequencies variations along the circular path. Frequency spectrum of the variation of end effector orientation obtained.

The Adept cycle responses are obtained using 2 motion laws with the same maximum velocity (v_max=2.88m/s):

• velocity trapeze;
• acceleration trapeze.

The active joints torques are greater in the case of acceleration trapeze than the velocity trapeze one because the maximum acceleration is higher (7.1g instead of 4.8g). However in the case of acceleration trapeze there are more torques oscillations at 180Hz during the approach and depart, so the robot control could be more difficult. These results are confirmed by the analysis of end effector position and orientation. The maximum values are during the travel motion but there are more vibrations during the approach and depart phases in the case of velocity trapeze than the acceleration trapeze one. The vibration frequency is about 165÷200Hz and it is the 13th-14th natural frequency. This is a vibration mode with a variation of end effector orientation, so the precision of the robot can be affected by connecting rod flexibility. Active joints torques in the cases of velocity trapeze (a) and acceleration trapeze (b) Frequency spectrum of the end effector x position in the cases of velocity trapeze (a) and acceleration trapeze (b).

# Conclusion

The workspace identification method shows that the robot workspace dimensions are independently controlled by the rocker arm length l_2 and connecting rod length l_4. The singularity regions increase if only a single geometrical parameter is modified, so both parameters l_2 and l_4 need to be increased  in order to avoid wide singularity regions. However the reduced inertia increases if the parameters l_2 and l_4 are both increased because the force arm is greater. A possible future development is to include the passive joints limits in the workspace identification method.

The robot vibration modes are with or without a variation of end effector orientation. The end effector vibrates along and around x and y directions because of the variation of end effector orientation. The response to the circular motion shows that the steady-state excitation is related to the natural frequencies variations and the excitation frequency is 3 times the rotating one. The Adept cycle responses show that the use of velocity trapeze instead of acceleration trapeze could causes:

• a difficult robot control because of torques oscillation;
• a low robot precision during depart and approach phases;
• a higher cycle time.

A possible future development is to include the friction force and to implement a simple robot control in order to analyze the consequence of different motion laws more.