Multibody Quantities

Quantities of interest in multibody dynamics have distinct types.

For example, a rotation matrix is denoted with `R`

and a position vector with `p`

. New quantities can be created by differentiation of an existing quantity (see Time Derivatives of Multibody Quantities).

Most quantities have a *reference* and *target*, either of which may be a frame, basis, or point, that specify how the quantity is defined. In computation, vectors are *expressed* in a specified basis (or frame) that associates a direction with each vector element. For example, the velocity of a point P moving in a reference frame F is a vector quantity with *target* point P and *reference* frame F. In typeset this symbol is written as \(^Fv^P\). Here v is the quantity type, the left superscript F is the reference, and the right superscript P is the target. In computation, this vector is *expressed* in a particular basis. By default, the assumed expressed-in frame is the same as the *reference* frame, so in this case, the assumed expressed-in frame is frame F's basis. Alternately, to use a different expressed-in frame, say W, typeset with the bracket notation: \([^Fv^P]_W\).

Kane/monogram notation was developed decades ago for kinematics and dynamics. It explicitly communicates the quantity being measured and how it is expressed. The typeset symbol \(^Fv^P\) is translated to monogram notation as `v_FP`

. The quantity type always comes first, then an underscore, then left and right superscripts. By default, the symbol `v_FP`

implies the vector is expressed in frame F. Alternately, to express in frame W, use the typeset \([^Fv^P]_W\) and monogram notation `v_FP_W`

(add a final underscore and expressed-in frame W). We adhere to this pattern for all quantities and it is quite useful once you get familiar with it. As a second example, consider the position vector of point Bcm (body B's center of mass) from point Bo (the origin of frame B), expressed in B. In explicit typeset, this is \([^{B_o}p^{B_{cm}}]_B \) whereas in implicit typeset this is abbreviated \(^Bp^{B_{cm}}\) (where the left-superscript B denotes Bo and the expressed-in frame is implied to be B). The corresponding monogram equivalents are `p_BoBcm_B`

and `p_BBcm`

, respectively.

Here are some more useful multibody quantities.

Quantity | Symbol | Typeset | Monogram | Meaningᵃ |
---|---|---|---|---|

Rotation matrix | R | \(^BR^C\) | `R_BC` | Frame C's orientation in frame B |

Position vector | p | \(^Pp^Q\) | `p_PQ` | Position vector from point P to point Q |

Transform/pose | X | \(^BX^C\) | `X_BC` | Frame C's rigid transform (pose) in frame B |

General Transform | T | \(^BT^C\) | `T_BC` | The relationship between two spaces – it may be affine, projective, isometric, etc. Every X_AB can be written as T_AB, but not every T_AB can be written as X_AB. |

Angular velocity | w | \(^B\omega^C\) | `w_BC` | Frame C's angular velocity in frame Bᵃ |

Velocity | v | \(^Bv^Q\) | `v_BQ` | Point Q's translational velocity in frame B |

Relative velocity | v | \(^Bv^{Q/P}\) | `v_B_PQ` | Point Q's translational velocity relative to point P in frame B |

Spatial velocity | V | \(^BV^{C}\) | `V_BC` | Frame C's spatial velocity in frame B (for point Co)ᵇ |

Relative spatial velocity | V | \(^BV^{C/D}\) | `V_B_DC` | Frame C's spatial velocity relative to frame D in frame B |

Angular acceleration | alpha | \(^B\alpha^C\) | `alpha_BC` | Frame C's angular acceleration in frame B |

Acceleration | a | \(^Ba^Q\) | `a_BQ` | Point Q's translational acceleration in frame B |

Relative acceleration | a | \(^Ba^{Q/P}\) | `a_B_PQ` | Point Q's translational acceleration relative to point P in frame B |

Spatial acceleration | A | \(^BA^{C}\) | `A_BC` | Frame C's spatial acceleration in frame B (for point Co)ᵇ |

Relative spatial acceleration | A | \(^BA^{C/D}\) | `A_B_DC` | Frame C's spatial acceleration relative to frame D in frame B |

Torque | t | \(\tau^{B}\) | `t_B` | Torque on a body (or frame) B |

Force | f | \(f^{P}\) | `f_P` | Force on a point P |

Spatial force | F | \(F^{B}\) | `F_B` | Spatial force (torque/force) on a body (or frame) Bᶜ |

Inertia matrix | I | \(I^{B/Bo}\) | `I_BBo` | Body B's inertia matrix about Bo |

Spatial inertia | M | \(M^{B/Bo}\) | `M_BBo` | Body B's spatial inertia about Boᵃ |

Spatial momentum | L | \(^BL^{S/P}\) | `L_BSP` | System S's spatial momentum about point P in frame B |

Spatial momentum | L | \(^BL^{S/Scm}\) | `L_BScm` | System S's spatial momentum about point Scm in frame B |

Kinetic energy | K | \(^BK^S\) | `K_BS` | System S's kinetic energy in frame B |

Jacobian wrt qᵈ | Jq | \([J_{q}^{{}^Pp^Q}]_E\) | `Jq_p_PQ_E` | Point Q's position Jacobian from point P in frame E wrt q (in means both measured-in and expressed-in) |

Jacobian wrt q̇ | Jqdot | \(J_{q̇}^{{}^Bv^Q}\) | `Jqdot_v_BQ` | Point Q's translational velocity Jacobian in frame B wrt q̇ |

Jacobian wrt v | Jv | \(J_{v}^{{}^Bv^Q}\) | `Jv_v_BQ` | Point Q's translational velocity Jacobian in frame B wrt v |

Jacobian wrt v | Jv | \(J_{v}^{{}^B\omega^C}\) | `Jv_w_BC` | Frame C's angular velocity Jacobian in frame B wrt v |

ᵃ In code, a vector has an expressed-in-frame which appears after the quantity.

Example: `w_BC_E`

is C's angular velocity in B, expressed in frame E, typeset as \([^B\omega^C]_E \). If not explicitly present, the expressed-in-frame defaults to the measured-in-frame. Hence, `w_BC`

is C's angular velocity **in** (measured-in) frame B, expressed in frame B (implied).

Similarly, an inertia matrix or spatial inertia has an expressed-in-frame.

Example: `I_BBo_E`

is body B's inertia matrix about Bo, expressed in frame E, typeset as \([I^{B/Bo}]_E\).

By convention, `I_BP`

is body B's inertia matrix about point P, expressed in B (implied).

For more information, see Spatial Mass Matrix (Spatial Inertia)

ᵇ In code, spatial velocity (or spatial acceleration) has an expressed-in-frame which appears after the quantity.

Example: `V_BC_E`

is frame C's spatial velocity in frame B, expressed in frame E and contains both `w_BC_E`

(described aboveᵃ) and v_BC_E (point Co's translational velocity in frame B, expressed in frame E). Reminder, a rigid body D's translational and spatial velocity are for point Do (the origin of D's body frame), not for Dcm (D's center of mass). See Frames and Bodies for more information.

ᶜ It is often useful to **replace** a set of forces on a body or frame B by an equivalent set with a force \(f^{Bo}\) (equal to the set's resultant) placed at Bo, together with a torque \(\tau\) equal to the moment of the set about Bo. A spatial force Fᴮᵒ containing \(\tau\) and \(f^{Bo}\) represents this replacement. Note: the spatial force Fᴮ is shorthand for Fᴮᵒ (i.e., the about-point is B's origin Bo). The monogram notation F_Bcm is useful when the about-point is Bcm (body B's center of mass).

ᵈ The Jacobian contains partial derivatives wrt (with respect to) scalars e.g., wrt q (generalized positions), or q̇, or v (generalized velocities). The example below shows the simplicity of Jacobian monogram: first is the Jacobian symbol (Jv), next is the kinematic quantity (v_BQ), last is an expressed-in frame (E).

Example: `Jv_v_BQ_E`

is `Jv`

(Jacobian wrt v), for `v_BQ`

(velocity in frame B of point Q), expressed in frame E.

**Advanced:** Due to rules of vector differentiation, explicit Jacobian monogram notation for `Jq`

(Jacobian wrt generalized positions q) requires an extra frame (e.g., `JBq`

is partial differentiation in frame B wrt q). Frequently, the partial-differentiation-in-frame B is identical to the expressed-in-frame E and a shorthand notation can be used.

Example: `Jq_p_PQ_E`

is `Jq`

(Jacobian **in** frame E wrt q), for `p_PQ`

(position vector from point P to point Q), expressed **in** frame E.

**Special relationship between position and velocity Jacobians:** When a point Q's position vector originates at a point Bo **fixed** in frame B and when there are no motion constraints (no relationships between q̇₁ ... q̇ₙ), \(\;[J_{q}^{{}^{Bo}p^Q}]_B = [J_{q̇}^{{}^Bv^Q}]_B\;\) i.e., `(Jq_p_BoQ_B = Jqdot_v_BQ_B)`

.

Monogram notation addresses frequently used kinematics. In a context-sensible way (and in collaboration with reviewers), extend notation to related concepts. Some examples are shown below.

Extended notation | Description (herein E is the vectors' expressed-in-frame) | |
---|---|---|

p_PQset_E | p_PQlist_E | Set of position vectors from a point P to each of the points in the set Q = {Q₀, Q₁, Q₂, ...} |

v_BQset_E | v_BQlist_E | Set of translational velocities in frame B for the set of points Q = {Q₀, Q₁, Q₂, ...} |

w_BCset_E | w_BClist_E | Set of angular velocities in frame B for the frames in the set C = {C₀, C₁, C₂, ...} |

vset_E | vlist_E | Set of generic vectors v = {v₀, v₁, v₂} expressed in frame E |

mesh_B | A mesh whose underlying vertices' positions are from Bo (frame B's origin), expressed in frame B | |

point_cloud_B | A point cloud whose underlying points' positions are from Bo (frame B's origin), expressed in frame B |

Next topic: Units of Multibody Quantities