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

scara robot

Robot link moment of inertia is relative to it's center of mass

Rotor moment of inertia is reflected to gearbox output shaft

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 $M_{i}$ for 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=1,...,3} is the mass of link 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} . Link 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 3} is a ball screw. The variable 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 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 $p$ are [mm/deg], while its user units are [m/rad].

The variables 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 L_1} and 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 L_2} are the lengths of links 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 1} and 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 2} . The variables 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 A_1} and 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 A_2} are respectively the distances of the exes of joints 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 1} and 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 2} to the centers of mass of links 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 1} and 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 2} .

The variable 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_i} for 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=1,...,3} is the moment of inertia of link 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} , 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 (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 J_i} for 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=1,...,4} is the moment of inertia of rotor 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} , reflected to the gearbox output shaft as follows. Let 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_{rotor,i}} be the moment of inertia of rotor 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} relative to a reference frame attached to the rotor's center of mass, about the rotor's axis of rotation. Let 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 GR_i} be the gear ratio of the gearbox or pulley system, then 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 J_i=I_{rotor,i} GR_i^2} .

The variables 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 M_0} and 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_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 (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 A_1^2 M_1+L_1^2 (M_2+M_3+M_0) + I_1 + J_1}	kg*m²
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 A_2^2 M_2 +L_2^2 (M_3+M_0)+ I_2}	kg*m²
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 J_2}	kg*m²
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 L_1 A_2 M_2 + L_1 L_2(M_3+M_0)}	kg*m²
5	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 M_3+M_0+J_3}	kg
6	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+M_0)}	kg*m/sec²
7	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_3+I_0}	kg*m²
8	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 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 (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 A_1^2 M_1+L_1^2 (M_2+M_3+M_0) + I_1 + J_1}	kg*m²
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 A_2^2 M_2 +L_2^2 (M_3+M_0)+ I_2}	kg*m²
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 L_1 A_2 M_2 + L_1 L_2(M_3+M_0)}	kg*m²
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 M_3+M_0}	kg
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_3+I_0}	kg*m²
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 J_2}	kg*m²
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 J_3}	kg
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 (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 (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 M_0 } , the mass of the payload
2	payloadInertia	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_0 } , the payload's moment of inertia 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, except for parameter No. 13.

The payload parameters are:

Number	Parameter	Comments
1	payloadMass	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 M_0} , the mass of the payload
2	payloadInertia	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_0} , the payload's moment of inertia relative to its center of mass
3	payloadLx	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 L_{0x}} , the distance to the center of mass from the 4th axis in the x direction
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_3+I_0+M_0 L_{0x}^2}	kg*m², includes the additional term 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 M_0 L_{0x}^2} according to parallel axis theorem

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 (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 \Theta_{AB}}	kg*m²
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 \cdot L_{AB} \cdot M_{AB}}	kg*m²/sec²
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 M_{BC}}	kg
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 \Theta_{BC}}	kg*m²
5	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 M_P}	kg
6	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 M_T}	kg
7	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 \Theta_T}	kg*m²
8	$\Theta _{T\phi }$	kg*m²
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 L_{TO}}	m
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 L_{TP}}	m
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 D}
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 C_r}
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 Fr_{max}}
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 R_{ext}}

Puma Robots

Dynamic Model 1

Puma robot

Description:

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 } - Gravity constant
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 m_{i} } - Mass of the i^th link
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 a_{i} } - length of the common normal between the i^th and i^th+1 joints
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 d_{i} } - offset along z axis between the i^th and i^th+1 joints
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 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 (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_{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 (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 (m_{3}+m_{4}+m_{5}+m_{6})(d_{2}+d_{3})^2+I_{2,xx}+I_{3,yy}+2m_{2}d_{2}l_{2,z}+m_{2}l_{2,y}^2+m_{3}l_{3,z}^2+2m_{3}(d_{2}+d_{3})*l_{3,z}+I_{4,zz}+I_{4,yy}+I_{6,zz}}	kg*m^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 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	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_{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 (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_{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 (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_{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 (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_{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 (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_{7} = m_{3}l_{3,y}^2+I_{3,xx}-I_{3,yy}+m_{4}l_{4,z}^2+2m_{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	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_{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} = 2m_{4}a_{5}l_{4,z}+2(m_{4}+m_{5}+m_{6})a_{3}d_{4}}	kg*m^2
11	$I_{11}=-2m_{2}l_{2,x}*l_{2,y}$	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	m_P	Payload mass [kg]
2	m_B	Balance mass [kg]
3	T_P	Payload mass center distance from the flange [mm]
4	T_B	Balance mass center distance from the (0,0) [mm]
5	I_R	Inertia of the payload around roll [kg*m²

Dynamic Models

Contents

Rotational Axes

Dynamic Model 1 - simple rotary axis

Dynamic Model 2 - horizontal crank-arm axis

Dynamic Model 3 - vertical crank-arm axis

Linear Axes

Dynamic Model 1 - horizontal axis

Dynamic Model 2 - vertical or tilted axis

Dynamic Model 3 - vertical axis with a spring

Traverse Arm Robots

Dynamic Model 1

Scara Robots

Dynamic Model 1

Dynamic Model 2

Dynamic Model 3

Dynamic Model 4

Dynamic Model 5

Delta Robots

Dynamic Model 1

Puma Robots

Dynamic Model 1

Dynamic Model 2 - Gravity

Galileo Spherical Robots (GSR)

Dynamic Model 1

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