Multi-Point Constraints (/MPC)

Gear type joints are more complex than other kinematic joints. They use the Lagrange Multiplier method and are compatible with all other Lagrange Multiplier kinematic conditions and incompatible with all classical kinematic conditions.

Three examples of these joints are explained:
  • Rotational gear type joint
  • Rack and pinion joint
  • Differential gear joint

Mass and inertia may be added to all nodes. MPC joints impose relations between nodes velocities. The MPC cannot be applied to the translational degrees of freedom of a node without mass or the rotational degrees of freedom of a node without inertia.

Rotational Gear Type Joint

This joint is used to impose a rotational velocity relation between input and output node as:


Figure 1. Rotational Type Joint
Translational velocities of gear joint nodes are constrained by a rigid link relation. For the rotational degrees of freedom, a scale factor is imposed between velocities of nodes N1 and N2, measured in their local coordinates. The corresponding constraint equations are:
α ( Δ ω 1 r 1 ) + ( Δ ω 2 r 2 ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aae WaaeaacqqHuoarcqaHjpWDdaWgaaWcbaGaaGymaaqabaGccqGHflY1 caWGYbWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaGaey4kaS YaaeWaaeaacqqHuoarcqaHjpWDdaWgaaWcbaGaaGOmaaqabaGccqGH flY1caWGYbWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaey ypa0JaaGimaaaa@4DF7@
Δ ω 1 s 1 = 0 ,    Δ ω 2 s 2 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeq yYdC3aaSbaaSqaaiaaigdaaeqaaOGaeyyXICTaam4CamaaBaaaleaa caaIXaaabeaakiabg2da9iaaicdacaGGSaGaaeiiaiaabccacqqHuo arcqaHjpWDdaWgaaWcbaGaaGOmaaqabaGccqGHflY1caWGZbWaaSba aSqaaiaaikdaaeqaaOGaeyypa0JaaGimaaaa@4C1C@ Δ ω 1 t 1 = 0 ,    Δ ω 2 t 2 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeq yYdC3aaSbaaSqaaiaaigdaaeqaaOGaeyyXICTaamiDamaaBaaaleaa caaIXaaabeaakiabg2da9iaaicdacaGGSaGaaeiiaiaabccacqqHuo arcqaHjpWDdaWgaaWcbaGaaGOmaaqabaGccqGHflY1caWG0bWaaSba aSqaaiaaikdaaeqaaOGaeyypa0JaaGimaaaa@4C1E@

Where, Δ ω 1 = ω 1 ω 0 ,    Δ ω 2 = ω 2 ω 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeq yYdC3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeqyYdC3aaSbaaSqa aiaaigdaaeqaaOGaeyOeI0IaeqyYdC3aaSbaaSqaaiaaicdaaeqaaO GaaiilaiaabccacaqGGaGaeuiLdqKaeqyYdC3aaSbaaSqaaiaaikda aeqaaOGaeyypa0JaeqyYdC3aaSbaaSqaaiaaikdaaeqaaOGaeyOeI0 IaeqyYdC3aaSbaaSqaaiaaicdaaeqaaaaa@4F08@ are relative rotational velocities of nodes N1 and N2 in respect of the rigid body rotational velocity.

Rack and Pinion Joint

This joint allows the rotational velocity of node to be transformed to a translational velocity as:


Figure 2. Rack and Pinion Type Joint
The constraint equations for these velocities are:(1)
Δ ω 1 = ω 1 ω 0 ,  Δ ω 2 = ω 2 ω 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeq yYdC3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeqyYdC3aaSbaaSqa aiaaigdaaeqaaOGaeyOeI0IaeqyYdC3aaSbaaSqaaiaaicdaaeqaaO GaaiilaiaabccacaqGGaGaeuiLdqKaeqyYdC3aaSbaaSqaaiaaikda aeqaaOGaeyypa0JaeqyYdC3aaSbaaSqaaiaaikdaaeqaaOGaeyOeI0 IaeqyYdC3aaSbaaSqaaiaaicdaaeqaaaaa@4F08@
(2)
α( Δ ω 1 s 1 )+( Δ V 2 s 2 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aae WaaeaacqqHuoarcqaHjpWDdaWgaaWcbaGaaGymaaqabaGccqGHflY1 caWGZbWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaGaey4kaS YaaeWaaeaacqqHuoarcaWGwbWaaSbaaSqaaiaaikdaaeqaaOGaeyyX ICTaam4CamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaiabg2 da9iaaicdaaaa@4D07@
(3)
α( Δ ω 1 t 1 )+( Δ V 2 t 2 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aae WaaeaacqqHuoarcqaHjpWDdaWgaaWcbaGaaGymaaqabaGccqGHflY1 caWG0bWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaGaey4kaS YaaeWaaeaacqqHuoarcaWGwbWaaSbaaSqaaiaaikdaaeqaaOGaeyyX ICTaamiDamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaiabg2 da9iaaicdaaaa@4D09@

Differential Gear Joint

This joint is used to impose rotational velocity relations between an input node and two output nodes as:


Figure 3. Differential Joint Type
The rotational velocities of a differential gear joint are constrained by the relations:(4)
α( Δ ω 1 r 1 )+( Δ ω 2 r 2 )+( Δ ω 3 r 3 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aae WaaeaacqqHuoarcqaHjpWDdaWgaaWcbaGaaGymaaqabaGccqGHflY1 caWGYbWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaGaey4kaS YaaeWaaeaacqqHuoarcqaHjpWDdaWgaaWcbaGaaGOmaaqabaGccqGH flY1caWGYbWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaey 4kaSYaaeWaaeaacqqHuoarcqaHjpWDdaWgaaWcbaGaaG4maaqabaGc cqGHflY1caWGYbWaaSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaa Gaeyypa0JaaGimaaaa@58BC@
(5)
α( Δ ω 1 s 1 )+( Δ ω 2 s 2 )+( Δ ω 3 s 3 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aae WaaeaacqqHuoarcqaHjpWDdaWgaaWcbaGaaGymaaqabaGccqGHflY1 caWGZbWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaGaey4kaS YaaeWaaeaacqqHuoarcqaHjpWDdaWgaaWcbaGaaGOmaaqabaGccqGH flY1caWGZbWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaey 4kaSYaaeWaaeaacqqHuoarcqaHjpWDdaWgaaWcbaGaaG4maaqabaGc cqGHflY1caWGZbWaaSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaa Gaeyypa0JaaGimaaaa@58BF@
(6)
α( Δ ω 1 t 1 )+( Δ ω 2 t 2 )+( Δ ω 3 t 3 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aae WaaeaacqqHuoarcqaHjpWDdaWgaaWcbaGaaGymaaqabaGccqGHflY1 caWG0bWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaGaey4kaS YaaeWaaeaacqqHuoarcqaHjpWDdaWgaaWcbaGaaGOmaaqabaGccqGH flY1caWG0bWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaey 4kaSYaaeWaaeaacqqHuoarcqaHjpWDdaWgaaWcbaGaaG4maaqabaGc cqGHflY1caWG0bWaaSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaa Gaeyypa0JaaGimaaaa@58C2@