97906 - Analysis and Synthesis of Biomechanical Systems

Learning outcomes

The student learns to apply the theoretical tools of applied mechanics for the analysis and modeling of biomechanical systems, with particular reference to the kinematic, static and dynamic problems of constrained body systems. In particular, he acquires skills in the modeling of biological systems using rigid and deformable elements; acquires the skills to develop complex models with proprietary computational codes and multibody software; extends this knowledge to the kinematic and dynamic design of mechanical medical devices, such as prostheses, orthoses and fixators.

Course contents

1. Mechanics applied to the analysis and synthesis of spatial and constrained mechanical systems for biomechanical applications
1.1. The human body as a spatial mechanism: rigid bodies, constraints, deformable elements, actuators
1.2. Orthopedic devices: mechanics and mechanisms applied to the human body


2. Spatial kinematic analysis of rigid bodies
2.1. Rigid bodies and reference systems
2.2. Coordinate systems for the description of the spatial motion of a body
2.3. Coordinate transformation and transformation matrix. Active and passive point of view.
2.4. Combining transformations with respect to fixed and moving references. Geometric and motion matrices.
2.5. Apparent translations.
2.6. Parameterization of position and orientation:
2.6.1. Grood and Suntay parameterization of the rotation matrix: direct and inverse problems. Position.
2.6.2. Euler-Rodrigues theorem and finite helical axis
2.6.3. Axis-angle parameterization of the rotation matrix: direct and inverse problems
2.6.4. Calculation of the finite helical axis. Advantages and applications of the helical finite axis. Motion axoids.
2.6.5. Euler parameters: direct and inverse problems.
2.6.6. Quaternions: definitions, properties, applications.
2.7. Velocity and acceleration of a rigid body
2.8. Matrix form of velocity and acceleration of a body. Change of reference system.
2.9. Derivatives of the rotation matrix. Calculation of angular velocity and acceleration.
2.10. Application to axis-angle parametrization and quaternions


3. Reference systems
3.1. Anatomical and technical
3.2. Definition based on points and surfaces
3.3. Transformations between multiple experimental data sets
3.4. Anatomical systems of the tibia and femur


4. Components of musculoskeletal models
4.1. Sensitivity of musculoskeletal systems and impact on joint forces
4.2. Joints and joint movements
4.2.1. Types of joints
4.2.2. Natural and charged motion
4.2.3. Synergies
4.3. Gdl and constraints
4.3.1. Number and type of dof in joints
4.3.2. Implicit or explicit constraints
4.3.3. Rigid and deformable constraints. Isometry of the ligaments
4.3.4. Isostatic and hyperstatic systems. Influence on kinematics and statics.
4.3.5. Types of hyperstaticity: joint geometric; kinematic motors; muscle implementation
4.3.6. Influence of medical devices on dofs and hyperstatics
4.3.7. Principle of virtual jobs. Influence of the shape of the constraints on the support reactions and on the resultant.
4.3.8. Methods to manage hyperstaticity
4.3.9. Rigid, elastic, rigid models with elastic bodies. Kinematic, static, dynamic models.
4.3.10. Model validation
4.4. Joint contacts
4.4.1. Mechanical behavior and applications of cartilage
4.4.2. Rigid and deformable models: kinematic, springs, spring mattress, FEM, Booleans
4.5. Ligaments
4.5.1. Role of the ligaments
4.5.2. Rigid models
4.5.3. Multifiber models: fiber maps, number of fibers and mechanical characteristics
4.5.4. Number of parameters
4.5.5. FEM models
4.6. Muscles
4.6.1. Generation of muscle forces
4.6.2. Model parameters
4.6.3. Over-implemented systems and co-contractions


5. Kinematic models
5.1. Analysis and synthesis of kinematic models. Topological and dimensional synthesis.
5.2. Methods for dimensional synthesis: geometric/anatomical, kinematic, functional, from literature
5.3. Analytical and optimization methods. Application examples on orthopedic devices: prostheses and fixators.
5.4. Methods for kinematic analysis:
5.4.1. Open and closed chain mechanisms
5.4.2. Movement parameters and geometric parameters
5.4.3. Closure equations: position, velocity, acceleration. 1 dof case. Principle of virtual jobs.
5.4.4. Methods of writing closure equations: representative vectors; transformation matrices; opening of the kinematic chain and scalar invariants
5.4.5. Impact on the number of equations and parameters
5.5. Analysis and synthesis of kinematic models with (generally) implicit constraints:
5.5.1. Rotary pair
5.5.2. Cardan joint
5.5.3. Spherical couple
5.5.4. Cam-synergies
5.6. Analysis and synthesis of kinematic models with explicit constraints
5.6.1. Equivalent mechanisms. Full and reduced 1-1 representation.
5.6.2. Calculation of dofs
5.6.3. Closure equations: position, velocity, acceleration
5.6.4. Articulated quadrilateral
5.6.5. 5-5
5.6.6. Spherical wrist
5.6.7. Shoulder mechanism
5.7. Applications to the design and analysis of medical devices
5.8. Kinematically equivalent constraints: rotoidal, spherical, double spherical, spherical+plane on plane, sphere on sphere, sphere on plane, rigid bodies, conjugated surfaces, zero work constraints


6. Inverse kinematics of over-actuated systems
6.1. Over-implemented mechanisms
6.2. Global or multibody optimization
6.3. Least squares based on Penrose pseudoinverse
6.4. Lagrange matrix and multiplier method

Readings/Bibliography

Material provided via "Virtuale". The lessons will also be recorded and made available to those enrolled in the course.

Teaching methods

The course consists of lectures and exercises. Some exercises will also be assigned to be carried out independently.

The lessons provide the theoretical basis for the analysis and synthesis of constrained systems, applied to biomechanical systems. Some physiological aspects are recalled (treated in other courses), to deepen the computational and modeling aspects. Some problems of kinematic and functional design of orthopedic devices are also addressed, also focusing on unconventional mechanisms and actuators.

The exercises put into practice some of the theoretical aspects covered in the lectures. We try to underline the importance of both the development of proprietary code and the use of existing software, up to show how to combine the two techniques by interfacing existing musculoskeletal models with proprietary models.

The assigned exercises, in addition to constituting a basis for evaluation, are designed to allow students to independently develop some concepts seen both in the theoretical lessons and during the exercises.

Assessment methods

The exam consists of two parts. In the first part, the exercises to be performed independently will be presented and commented on. The results obtained and the correctness of the procedure will be evaluated.

The second part is made up of theoretical questions on the topics of the course. Both general knowledge and computational aspects will be verified.

Teaching tools

Material prepared by the teacher on "Virtuale" will be provided. The lessons will also be recorded and made available to those enrolled in the course.

Furthermore, free software will be used to carry out the exercises. Further information will be provided during the course and on "Virtuale".

Office hours

See the website of Nicola Sancisi