Dynamic Models

This page gives an overview over all implemented dynamic models.

General considerations

• Friction is handled on axis basis. The parameters for friction are set for each axis separately.
• Torque (Force) is always expressed in [Nm] ([N])
  

Rotational Axes

Dynamic Model 1 - simple rotary axis

Number	Parameter	Comments
1	$I$	Total moment of inertia around the rotation axis of the moved part

Model equation

$T=(I+I_{payload})\cdot acc$

Dynamic Model 2 - horizontal crank-arm axis

Horizontal crank-arm axis

Number	Parameter	Comments
1	$I$	Total moment of inertia around the rotation axis of the moved part
2	$L^{2}$	Square of length of crank arm (axis to payload)

Model equation

$T=(I+I_{payload}+L^{2}\cdot M_{payload})\cdot acc$

Dynamic Model 3 - vertical crank-arm axis

Vertical crank-arm axis

Number	Parameter	Comments
1	$I$	Total moment of inertia around the rotation axis of the moved part
2	$L^{2}$	Square of length of crank arm (axis to payload)
3	$M\cdot g\cdot A$	Mass (without payload) * Gravity * Distance to center of mass
4	$g\cdot L$	Gravity * Distance to Payload

Model equation

$T=(I+I_{payload}+L^{2}\cdot M_{payload})\cdot acc-(M\cdot g\cdot A+M_{payload}\cdot g\cdot L)\cdot \sin(pos)$

Linear Axes

Dynamic Model 1 - horizontal axis

Horizontal linear axis

Number	Parameter	Comments
1	$M$	Total mass of the moved part.

Model equation

$T=(M+M_{payload})\cdot acc$

Dynamic Model 2 - vertical or tilted axis

Vertical linear axis

Number	Parameter	Comments
1	$M$	Total mass of the moved part.
2	$M\cdot g\cdot \cos(\alpha )$	Constant force due to gravity.
3	$g\cdot \cos(\alpha )$	Gravity coefficient used to consider payload mass. (g = 9.80665)

Model equation

$T=(M+M_{payload})\cdot acc+M\cdot g\cdot \cos(\alpha )+M_{payload}\cdot g\cdot \cos(\alpha )$

Dynamic Model 3 - vertical axis with a spring

Vertical linear axis with a spring

Number	Parameter	Comments
1	$M$	Total mass of the moved part. [kg]
2	$K$	The stiffness constant of the spring. [kg/s^2]
3	$K\cdot X_{0}$	The stiffness constant times the relaxation position of the spring. [kg*m/s^2]

Model equation

$T=(M+M_{payload})\cdot (acc+g)+K\cdot (X-X_{0})$

Traverse Arm Robots

Dynamic Model 1

Traverse Arm robot

Number	Parameter	Comments
1	$M_{1}+M_{2}$
2	$A_{2}\cdot M_{2}$
3	$A_{2}^{2}\cdot M_{2}+I_{2}$
4	$M_{3}$
5	$g\cdot M_{3}$
6	$I_{4}$
7	$J_{4}$

Scara Robots

Dynamic Model 1

scara robot

Number	Parameter	Comments
1	$L_{1}^{2}\cdot M_{2}+I_{1}$
2	$A_{2}^{2}\cdot M_{2}+I_{2}$
3	$J_{2}$
4	$L_{1}\cdot A_{2}\cdot M_{2}$
5	$M_{3}$
6	$g\cdot M_{3}$
7	$I_{4}$
8	$J_{4}$

Delta Robots

Dynamic Model 1

Delta robot

Number	Parameter	Comments
1	$\Theta _{AB}$	kg*m2
2	$g\cdot L_{AB}\cdot M_{AB}$	kg*m2/sec2
3	$M_{BC}$	kg
4	$\Theta _{BC}$	kg*m2
5	$M_{P}$	kg
6	$M_{T}$	kg
7	$\Theta _{T}$	kg*m2
8	$\Theta _{T\phi }$	kg*m2
9	$L_{TO}$	m
10	$L_{TP}$	m
11	$D$
12	$C_{r}$
13	$Fr_{max}$
14	$R_{ext}$

Puma Robots

Dynamic Model 1

Puma robot

Description:

• $g$ - Gravity constant
• $m_{i}$ - Mass of the i^th link
• $a_{i}$ - length of the common normal between the i^th and i^th+1 joints
• $d_{i}$ - offset along z axis between the i^th and i^th+1 joints
• $l_{i}$ - The distance from the i^th joint to the center of mass of the i^th link

Number	Parameter	Comments
1	$I_{1}=I_{1,zz}+m_{1}*l_{1,y}^{2}+m_{2}*d_{2}^{2}+(m_{4}+m_{5}+m_{6})*a_{3}^{2}+m_{2}*l_{2,z}^{2}+$ $(m_{3}+m_{4}+m_{5}+m_{6})*(d_{2}+d_{3})^{2}+I_{2,xx}+I_{3,yy}+2*m_{2}*d_{2}*l_{2,z}+m_{2}*l_{2,y}^{2}+m_{3}*l_{3,z}^{2}+2*m_{3}*(d_{2}+d_{3})*l_{3,z}+I_{4,zz}+I_{4,yy}+I_{6,zz}$	kg*m^2
2	$I_{2}=I_{2,zz}+m_{2}*(l_{2,x}^{2}+l_{2,y}^{2})+(m_{3}+m_{4}+m_{5}+m_{6}*a_{2}^{2}$	kg*m^2
3	$I_{3}=-I_{2,xx}+I_{2,yy}+(m_{3}+m_{4}+m_{5}+m_{6})*a_{2}^{2}+m_{2}*l_{2,x}^{2}-m_{2}*l_{2,y}^{2}$	kg*m^2
4	$I_{4}=m_{2}*l_{2,x}*(d_{2}+l_{2,z})+m_{3}*a_{2}*l_{3,z}+(m_{3}+m_{4}+m_{5}+m_{6})*a_{2}*(d_{2}+d_{3})$	kg*m^2
5	$I_{5}=-m_{3}*a_{2}*l_{3,y}+(m_{4}+m_{5}+m_{6})*a_{2}*d_{4}+m_{4}*a_{2}*l_{4,z}$	kg*m^2
6	$I_{6}=I_{3,zz}+m_{3}*l_{3,y}^{2}+m_{4}*a_{3}^{2}+m_{4}*(d_{4}+l_{4,z})^{2}+I_{4,yy}+m_{5}*a_{3}^{2}+m_{5}*d_{4}^{2}+I_{5,zz}+m_{6}*a_{3}^{2}+m_{6}*d_{4}^{2}+m_{6}*l_{6,z}^{2}+I_{6,xx}$	kg*m^2
7	$I_{7}=m_{3}*l_{3,y}^{2}+I_{3,xx}-I_{3,yy}+m_{4}*l_{4,z}^{2}+2*m_{4}*d_{4}*l_{4,z}+(m_{4}+m_{5}+m_{6})*(d_{4}^{2}-a_{3}^{2})+I_{4,yy}-I_{4,yy}+I_{5,zz}-I_{5,yy}+m_{6}*l_{6,z}^{2}-I_{6,zz}+I_{6,xx}$	kg*m^2
8	$I_{8}=-m_{4}*(d_{2}+d_{3})*(d_{4}+l_{4,z})-(m_{3}+m_{6})*(d_{2}+d_{3})*d_{4}+m_{3}*l_{3,y}*l_{3,z}+m_{3}*(d_{2}+d_{3})*l_{3,y}$	kg*m^2
9	$I_{9}=m_{2}*l_{2,y}*(d_{2}+l_{2,z})$	kg*m^2
10	$I_{10}=2*m_{4}*a_{5}*l_{4,z}+2*(m_{4}+m_{5}+m_{6})*a_{3}*d_{4}$	kg*m^2
11	$I_{11}=-2*m_{2}*l_{2,x}*l_{2,y}$	kg*m^2
12	$I_{12}=(m_{4}+m_{5}+m_{6})*a_{2}*a_{3}$	kg*m^2
13	$I_{13}=(m_{4}+m_{5}+m_{6})*a_{3}*(d_{2}+d_{3})$	kg*m^2
14	$I_{14}=I_{4,zz}+I_{5,yy}+I_{6,zz}$	kg*m^2
15	$I_{15}=m_{6}*d_{4}*l_{6,z}$	kg*m^2
16	$I_{16}=m_{6}*a_{2}*l_{6,z}$	kg*m^2
17	$I_{17}=I_{5,zz}+I_{6,xx}+m_{6}*l_{6,z}^{2}$	kg*m^2
18	$I_{18}=m_{6}*(d_{2}+d_{3})*l_{6,z}$	kg*m^2
19	$I_{19}=I_{4,yy}-I_{4,xx}+I_{5,zz}-i_{5,yy}+m_{6}*l_{6,z}^{2}+I_{6,xx}-I_{6,zz}$	kg*m^2
20	$I_{20}=I_{5,yy}-I_{5,xx}-m_{6}*l_{6,z}^{2}+I_{6,zz}-I_{6,xx}$	kg*m^2
21	$I_{21}=I_{4,xx}-I_{4,yy}+I_{5,xx}-I_{5,zz}$	kg*m^2
22	$I_{22}=m_{6}*a_{3}*l_{6,z}$	kg*m^2
23	$I_{23}=I_{6,zz}$	kg*m^2

Dynamic Model 2 - Gravity

This dynamic model is for cases where the robot moves very slowly.
In such cases, the accelerations and velocities of the joints of the robot have little effect on the joints torques. The joints torques are mainly affected by gravity and friction.
This model includes only the gravity and friction part of the PUMA robot dynamic model.

Number	Parameter	Comments
1	$g(m_{2}l_{2}+a_{2}(m_{3}+m_{4}+m_{5}+m_{6}))$	kg*m^2/s^2
2	$g(m_{3}l_{3,y}+a_{3}(m_{4}+m_{5}+m_{6}))$	kg*m^2/s^2
3	$g(m_{3}l_{3,y}+m_{4}(l_{4}+d_{4}(m_{5}+m_{6}))$	kg*m^2/s^2
4	$gl_{56}(m_{5}+m_{6})$	kg*m^2/s^2

Galileo Spherical Robots (GSR)

Dynamic Model 1

Galileo robot

Number	Parameter	Comments
1	mP	Payload mass [kg]
2	mB	Balance mass [kg]
3	TP	Payload mass center distance from the flange [mm]
4	TB	Balance mass center distance from the (0,0) [mm]
5	IR	Inertia of the payload around roll [kg*m2