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	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I}	Total moment of inertia around the rotation axis of the moved part

Model equation

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T=(I+I_{payload})\cdot acc}

Dynamic Model 2 - horizontal crank-arm axis

Horizontal crank-arm axis

Number	Parameter	Comments
1	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I}	Total moment of inertia around the rotation axis of the moved part
2	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L^{2}}	Square of length of crank arm (axis to payload)

Model equation

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I}	Total moment of inertia around the rotation axis of the moved part
2	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L^{2}}	Square of length of crank arm (axis to payload)
3	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle M\cdot g\cdot A}	Mass (without payload) * Gravity * Distance to center of mass
4	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g\cdot L}	Gravity * Distance to Payload

Model equation

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle M}	Total mass of the moved part.

Model equation

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T=(M+M_{payload})\cdot acc}

Dynamic Model 2 - vertical or tilted axis

Vertical linear axis

Number	Parameter	Comments
1	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle M}	Total mass of the moved part.
2	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle M\cdot g\cdot \cos(\alpha )}	Constant force due to gravity.
3	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g\cdot \cos(\alpha )}	Gravity coefficient used to consider payload mass. (g = 9.80665)

Model equation

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle M}	Total mass of the moved part. [kg]
2	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle K}	The stiffness constant of the spring. [kg/s^2]
3	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle K\cdot X_{0}}	The stiffness constant times the relaxation position of the spring. [kg*m/s^2]

Model equation

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle M_{1}+M_{2}}
2	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A_{2}\cdot M_{2}}
3	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A_{2}^{2}\cdot M_{2}+I_{2}}
4	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle M_{3}}
5	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g\cdot M_{3}}
6	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I_{4}}
7	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle J_{4}}

Scara Robots

scara robot

The dynamic equations of the robot are expressed at the outputs of the gearboxes attached to the actuation motors. Therefore, the torques, joint positions, velocities, and accelerations are those of the gearbox output shafts.

In the following models, the variable Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle M_{i}} for $i=1,...,3$ is the mass of link Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i} . Link Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3} is a ball screw. The variable Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p} is the lead of the ball screw, i.e. the linear distance traveled for each complete turn of the screw. The internal units of Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p} are [mm/deg], while its user units are [m/rad].

The variables Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L_{1}} and Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L_{2}} are the lengths of links $1$ and Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2} . The variables Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A_{1}} and Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A_{2}} are respectively the distances of the exes of joints $1$ and Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2} to the centers of mass of links $1$ and Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2} .

The variable Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I_{i}} for $i=1,...,3$ is the moment of inertia of link Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i} , relative to a reference frame attached to the link's center of mass, and about the link's axis of rotation.

Additionally, Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle J_{i}} for Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i=1,...,4} is the moment of inertia of rotor Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i} , reflected to the gearbox output as follows. Let Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I_{rotor,i}} be the moment of inertia of rotor Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i} relative to a reference frame attached to the rotor's center of mass, about the rotor's axis of rotation. Let Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle GR_{i}} be the gear ratio of the gearbox or pulley system, then Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle J_{i}=I_{rotor,i}GR_{i}^{2}} .

The variables Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle M_{0}} and Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I_{0}} are respectively the payload mass and payload moment of inertia relative to its center of mass. If no object or gripper is attached to the robot, the value of these variables is zero.

 

Dynamic Model 1

For non-coupled SCARA robots (axis 3 and 4 are not coupled) and for concentric payloads (concentric with axis 4).

Number	Parameter	Comments
1	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A_{1}^{2}M_{1}+L_{1}^{2}(M_{2}+M_{3}+M_{0})+I_{1}+J_{1}}	kg*m2
2	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A_{2}^{2}M_{2}+L_{2}^{2}(M_{3}+M_{0})+I_{2}}	kg*m2
3	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle J_{2}}	kg*m2
4	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L_{1}A_{2}M_{2}+L_{1}L_{2}(M_{3}+M_{0})}	kg*m2
5	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle M_{3}+M_{0}+J_{3}}	kg
6	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g(M_{3}+M_{0})}	kg*m/sec2
7	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I_{3}+I_{0}}	kg*m2
8	$J_{4}$	kg*m2

 

 

Dynamic Model 2

For coupled SCARA robots (axis 3 and 4 are coupled) and for concentric payloads (concentric with axis 4).

Number	Parameter	Comments
9	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A_{1}^{2}M_{1}+L_{1}^{2}(M_{2}+M_{3}+M_{0})+I_{1}+J_{1}}	kg*m2
10	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A_{2}^{2}M_{2}+L_{2}^{2}(M_{3}+M_{0})+I_{2}}	kg*m2
11	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L_{1}A_{2}M_{2}+L_{1}L_{2}(M_{3}+M_{0})}	kg*m2
12	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle M_{3}+M_{0}}	kg
13	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I_{3}+I_{0}}	kg*m2
14	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle J_{2}}	kg*m2
15	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle J_{3}}	kg
16	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (M_{3}+M_{0})p^{2}+J_{4}}	kg*m2

 

The payload parameters for Dynamic Model 1 and Dynamic Model 2 are:

Number	Parameter	Comments
1	payloadMass	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle M_{0}} , the mass of the payload
2	payloadInertia	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I_{0}} , the Inertia of the payload relative to its center of mass

 

Dynamic Model 3

For coupled SCARA robots (axis 3 and 4 are coupled) and for non-concentric payloads (non-concentric with axis 4).
The dynamic parameters are the same as model 2.

The payload parameters are:

Number	Parameter	Comments
1	payloadMass	The mass of the payload
2	payloadInertia	The Inertia of the payload relative to its center of mass
4	payloadLx	The distance to the center of mass from the 4th axis in the x direction

When using identification with this model, all of the payload parameters can be found.

 

Dynamic Model 4

Same as Dynamic Model 3.
This model is used in identification process in order to identify the payloadMass parameter only, by moving only joint number 3.

 

Dynamic Model 5

Same as Dynamic Model 3.
This model is used in identification process in order to identify the payloadInertia and payloadLx parameters, by moving only joint number 4 and very small movements in joints 1 and 2.

Delta Robots

Dynamic Model 1

Delta robot

Number	Parameter	Comments
1	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Theta _{AB}}	kg*m2
2	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g\cdot L_{AB}\cdot M_{AB}}	kg*m2/sec2
3	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle M_{BC}}	kg
4	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Theta _{BC}}	kg*m2
5	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle M_{P}}	kg
6	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle M_{T}}	kg
7	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Theta _{T}}	kg*m2
8	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Theta _{T\phi }}	kg*m2
9	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L_{TO}}	m
10	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L_{TP}}	m
11	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D}
12	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle C_{r}}
13	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Fr_{max}}
14	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle R_{ext}}

Puma Robots

Dynamic Model 1

Puma robot

Description:

• Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g} - Gravity constant
• Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle m_{i}} - Mass of the i^th link
• Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a_{i}} - length of the common normal between the i^th and i^th+1 joints
• Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d_{i}} - offset along z axis between the i^th and i^th+1 joints
• Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle l_{i}} - The distance from the i^th joint to the center of mass of the i^th link

Number	Parameter	Comments
1	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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}+} Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (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	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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	Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle I_{9} = m_{2}*l_{2,y}*(d_{2}+l_{2,z})}	kg*m^2
10	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle I_{11} = -2*m_{2}*l_{2,x}*l_{2,y}}	kg*m^2
12	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle I_{12} = (m_{4}+m_{5}+m_{6})*a_{2}*a_{3}}	kg*m^2
13	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle I_{13} = (m_{4}+m_{5}+m_{6})*a_{3}*(d_{2}+d_{3})}	kg*m^2
14	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle I_{14} = I_{4,zz}+I_{5,yy}+I_{6,zz}}	kg*m^2
15	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle I_{15} = m_{6}*d_{4}*l_{6,z}}	kg*m^2
16	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle I_{16} = m_{6}*a_{2}*l_{6,z}}	kg*m^2
17	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle I_{17} = I_{5,zz}+I_{6,xx}+m_{6}*l_{6,z}^2}	kg*m^2
18	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle I_{18} = m_{6}*(d_{2}+d_{3})*l_{6,z}}	kg*m^2
19	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle I_{20} = I_{5,yy}-I_{5,xx}-m_{6}*l_{6,z}^2+I_{6,zz}-I_{6,xx}}	kg*m^2
21	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle I_{21} = I_{4,xx}-I_{4,yy}+I_{5,xx}-I_{5,zz}}	kg*m^2
22	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle I_{22} = m_{6}*a_{3}*l_{6,z}}	kg*m^2
23	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g(m_{2}l_{2}+a_{2}(m_{3}+m_{4}+m_{5}+m_{6}))}	kg*m^2/s^2
2	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g(m_{3}l_{3,y}+a_{3}(m_{4}+m_{5}+m_{6}))}	kg*m^2/s^2
3	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g(m_{3}l_{3,y}+m_{4}(l_{4}+d_{4}(m_{5}+m_{6}))}	kg*m^2/s^2
4	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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