Davies Method

Davies Method is a method for computing the differential kinematics and statics of a general mechanical system using Graph Theory, Screw Theory and Plucker Coordinates.

Start with a Closed Kinematic Chain System (can be used with a Open Kinematic Chain System but it requires virtual links and joints) then follow the procedure

Enumerate the joints and links

Use letters for the joints and numbers for the links. The earth or support should be number 0 or 1 (when there is no 0).

Build a Coupling Graph

Build a Tree Graph by transforming the coupling graph into a tree.

Account for the number os branches (k) and the number of strings (v) in the final tree.

Evalute the degrees of freedom (f) and constraints (c) of each joint.

Kinematic Analysis

Build the Motion Graph

Build the Networks Graph

Build the Circuits Matrix (B)

Define the twists of the edges by Twist of a Joint.

Build the Direct Twists Matrix (M_D)

Acho esse passo desnecessário, dá pra construir M_N sem M_D.

Build the Motion Network Matrix (M_N)

Define the Vector of Motion Parameters (\(\varphi_M\))

The Kinematics realtions can be found by:

\[M_N \ \varphi_M = 0\]

Statics Analysis

Build the Action Graph

Build the Cut-Set Matrix (Q_a)

Define the wrenches of the edges by Wrench of a Joint.

Build the Direct Wrenches Matrix (A_D)

Acho esse passo desnecessário, dá pra construir A_N sem A_D.

Build the Action Network Matrix (A_N)

Define the Vector of Action Parameters (\(\varphi_A\))

The Statics relations can be found by:

\[A_N \ \varphi_A = 0\]

Coupling Between Kinematics and Statics

There may be a coupling between kinematics and statics, much like in the form of a viscous friction.

Solving the System

Build the Vector of System Parameters (\(\varphi\))

Build the Coupled Equations Matrix (\(C_{T\omega}\))

Build the System Matrix (\(M_S\))

The combined equation for Kinematics and Statics can be found by:

\[M_S \ \varphi = 0\]

Arrange the known parameters to the other side

Localize the known variables

Extract the respective column of the known variables and negate it to the other side of the equation.

Now you have:

• \(\varphi^*:\) \(\varphi\) without the extracted variables.

• \(M_S^* :\) \(M_S\) without the columns related to the extracted variables.

• \(b_S:\) Other side of the equation with the extracted varibles combined.

Solve the resulting linear system to compute the Kinematics and Statics

\[M_S^* \ \varphi^* = b_s\]